# Zames-Falb Multipliers: don’t panic

Matthew C. Turner
Cyber Physical Systems Research Group,
Electronics and Computer Science,
University of Southampton,
Southampton,
SO17 1BJ, UK.
Email<EMAIL_ADDRESS>
###### Abstract

Zames-Falb multipliers are mathematical constructs which can be used to prove
stability of so-called Lur’e systems: systems that consist of a feedback
interconnection of a linear element and a static nonlinear element. The main
advantage of Zames-Falb multipliers is that they enable “passivity”-like
results to be obtained but with a level of conservatism much lower than _pure_
passivity results. However, some of the papers describing the development of
the Zames-Falb multiplier machinery are somewhat abstruse and not entirely
clear. This article attempts to provide a relatively simple construction of
Zames and Falb’s main results which will hopefully be understandable to most
graduate-level control engineers.

##  1 Introduction

Figure 1:  Lur’e system: a feedback interconnection of a linear system and a
static nonlinearity. In this article positive feedback is assumed.

Absolute stability theory is the branch of control theory concerned with
establishing stability of a feedback interconnection of the type depicted in
Figure 1. It is routinely taught in graduate courses on nonlinear control, but
often only the popular techniques of the Circle and Popov Criteria are
covered, both of which assume the nonlinearity satisfies so-called sector
bounds. This is partly because the Circle and Popov Criteria have a nice
Lyapunov interpretation [17] and partly because they have an appealing
graphical form (in the single-input-single-output case). They can however be
quite conservative, sometimes not providing stability guarantees when the
feedback interconnection is, in fact, stable.

When one is looking for less conservative results than the Circle and Popov
Criteria, one’s attention is inevitably drawn to the so-called “Zames-Falb”
multipliers which, in a sense, generalise the Circle and Popov Criteria to the
case when the nonlinearity is _slope-restricted_ : a tighter condition than
sector-boundedness. Numerous papers [37, 5, 11] have demonstrated the
superiority of the Zames-Falb multipliers over the Circle/Popov Criteria in
terms of lower levels of conservatism. However, a treatment of Zames-Falb
multipliers remains quite difficult to incorporate into a typical UK-style
Master’s course.

This article is an attempt to address this problem and has been initiated by
student questions on further reading in my graduate-level Nonlinear Control
course, and also by questions I have been asked since I published the articles
[36, 35] which were based on Zames-Falb multipliers. Central to many of these
questions were some fundamental issues with the understanding of Zames-Falb
multipliers. While there are several excellent texts treating Zames-Falb
multipliers and related concepts ([10, 41]) and several papers which also
attempt to provide more background for the reader ([15]), most of these texts
are somewhat terse and difficult for the typical graduate control engineer to
understand. In addition, the term “Zames-Falb Multiplier” seems to be rather
off-putting and a cause of anxiety in the student of absolute stability. My
aim here is to provide a fairly simple and easy to follow introduction to
Zames-Falb multipliers which prioritises clarity over succinctness; hence the
title. I hope this is of use to readers and fills a niche in the literature on
Zames-Falb multipliers.

###  1.1 Scope of the article

This article is entirely different from the review paper [6] which provides
more of an overview of the state-of-the-art, and really concentrates on
searches and properties of Zames-Falb multipliers. Instead, this article
attempts to give a complete but understandable proof of the main results of
Zames-Falb, introduced in their paper [44]. The article also tries to give
some background to the work of Zames-Falb and towards the end touches on the
problems associated with _actually finding_ or _computing_ Zames-Falb
multipliers - this is biased towards my own research.

This article is _not_ a review of absolute stability in general, nor does it
cover Zames’ other significant contribution on conic sector stability analysis
[42, 43] (see also [27, 3, 34]).

###  1.2 Style of article

††margin:  _“I like the cover,” he said. “Don’t Panic. It’s the first helpful
or intelligible thing anybody’s said to me all day.”_

I have tried to keep the article as readable as possible but inevitably there
is a certain amount of notation and some non-trivial concepts which need
introducing in order to establish the results. Most of the notation follows
that in [46]. The article tries to use simple terminology and language where
possible and the treatment of some concepts (such as causality and well-
posedness) are deliberately light-touch. There is a certain amount of
whimsicality in the writing which is there to, hopefully, ensure the article
is fairly light. The quotes from Reference [1] are hopefully appropriate and
in keeping with the nature of the article.

##  2 Systems under consideration

Zames-Falb multipliers were developed to tackle the absolute stability
problem. The best way to understand the absolute stability problem is from the
diagram in Figure 1, which depicts a feedback interconnection containing a
linear-time-invariant (LTI) system $\mathbf{G}$ and a static feedback
nonlinearity $\phi(\cdot)$ \- this interconnection is sometimes called a
Lur’e/Lurie/Lurye system. The linear system $\mathbf{G}$ can be thought of as
a linear operator, with an associated transfer function $G(s)$ and impulse
response $g(t)$. It is also possible to interpret it as a state-space system,
although this is not required here.

The nonlinearity $\phi(.)$ here is considered to be _static_ (sometimes also
called memoryless, it has no dynamics) and _scalar valued_ , that is

$\phi(\cdot):\mathbb{R}\mapsto\mathbb{R}$

The absolute stability problem involves finding conditions which guarantee
stability of the interconnection for _classes_ of nonlinearity i.e. we seek
stability for all $\phi\in\mathcal{N}$ where $\mathcal{N}$ is some class of
nonlinearity. In the Circle and Popov Criteria the class of nonlinearities
$\mathcal{N}$ is the class of sector bounded nonlinearities, but different
classes - introduced shortly - will be of interest as well.

Remark – Robust Control. The absolute stability problem is, in essence, a form
of _robust control_ problem where the nonlinearity $\phi(\cdot)$ represents
the _uncertainty_ in the system. In the robust control literature,
$\phi(\cdot)$ is often labelled $\Delta$ where $\Delta$ represents a broader
class of uncertainty - see, for example, [21]. $\square$

###  2.1 Stability

In this article we are concerned with _input-output_ stability and more
specifically with so-called $\mathcal{L}_{2}~{}$stability. Central to this
definition is the so-called $\mathcal{L}_{2}~{}$space and its extension

$\displaystyle\mathcal{L}_{2}$
$\displaystyle:=\left\\{x(t)~{}:~{}\int_{0}^{\infty}\|x(t)\|^{2}~{}dt<\infty\right\\}$
(1) $\displaystyle\mathcal{L}_{2e}$
$\displaystyle:=\left\\{x(t)~{}:~{}\int_{0}^{T}\|x(t)\|^{2}~{}dt<\infty\quad\forall\quad
0\leq T<\infty\right\\}$ (2)

The space $\mathcal{L}_{2}~{}$represents all _square integrable_ signals and
roughly speaking represents most _well behaved signals_. The space
$\mathcal{L}_{2e}$ represents most signals which we may have to deal with and
it is important since, from an operator perspective we represent systems as
mappings between $\mathcal{L}_{2e}$ spaces. For example an LTI system
$\mathbf{G}$ can be considered as a mapping between these spaces, that is

$\mathbf{G}:\mathcal{L}_{2e}\rightarrow\mathcal{L}_{2e}$

From an operator perspective, the nonlinearity $\phi$ also maps
$\mathcal{L}_{2e}$ to $\mathcal{L}_{2e}$.

With reference to Figure 1 stability is then defined with respect to the
signal space $\mathcal{L}_{2}$. Specifically, given a LTI system $\mathbf{G}$
and a static nonlinearity $\phi(\cdot)$:

The feedback system in Figure 1 is said to be _stable_ if the signals
$u(t),y(t)\in\mathcal{L}_{2}$ for all exogenous signals
$r_{1}(t),r_{2}(t)\in\mathcal{L}_{2}$.

In other words:

_“Well behaved inputs lead to well behaved outputs”_

Good references on $\mathcal{L}_{2}~{}$stability are [10, 13] and [16]. Here
it is sufficient to understand stability as defined above.

A standing assumption throughout the paper is that the LTI element in Figure 1
is stable and causal: its transfer function contains only open-left-half-plane
poles. Such a system is (again following the robust control literature)
labelled $\mathcal{RH}_{\infty}$. Specifically, the assumption

$\mathbf{G}\in\mathcal{RH}_{\infty}$

is made throughout the paper. The assumption made on the Zames-Falb
multiplier, introduced shortly, will not be as strict. In this case, the only
requirement is that the multiplier is bounded on the imaginary axis111Here,
the multiplier is assumed to be rational, although this is not actually
_required_.. This is often denoted as

$\mathbf{M}\in\mathcal{RL}_{\infty}$

and can be interpreted as $\mathbf{M}$ consisting of both causal and anti-
causal elements:

$\mathbf{M}=\mathbf{M}_{c}+\mathbf{M}_{a}$

where $\mathbf{M}_{c}\in\mathcal{RH}_{\infty}$ and
$\mathbf{M}_{a}\in\mathcal{RH}_{\infty}^{\perp}$ i.e. $\mathbf{M}_{a}$
contains poles only in the open right half complex plane. For much of this
article, one may essentially ignore this detail, but it is stressed that for
searches there is often a useful advantage in allowing the multiplier to have
an anti-causal component [5]. On a first read this subtlety can probably be
ignored.

††margin:  _“Just believe everything I tell you, and it will all be very, very
simple.”
“Ah, well, I’m not sure I believe that”_

###  2.2 Nonlinearities

Figure 2:  Nonlinearities: the left-hand diagram shows sector $[0,1]$ and an
example nonlinearity; the right hand diagram shows an example slope-restricted
nonlinearity.

The classical Circle and Popov criteria both consider so-called sector bounded
nonlinearities in which the nonlinearity $\phi(.)$ satisfies the _sector
bounds_

$\displaystyle\alpha_{1}\leq\frac{\phi(\sigma)}{\sigma}\leq\alpha_{2}\quad\forall\sigma\in\mathbb{R}$
(3)

where $\alpha_{1}<\alpha_{2}$ are two real, not necessarily positive
constants. The short hand

$\phi\in{\rm Sector}[\alpha_{1},\alpha_{2}]$

is used to indicate a nonlinearity satisfying inequality (3). Many
nonlinearities satisfy sector bounds: the saturation and deadzone functions
being two common ones. The concept of a sector is illustrated in Figure 2;
note often the lower sector bound is taken as zero (something which can be
achieved with a loopshift - see [17], Chapter 10).

The main problem with the sector bounds is that many nonlinearities inhabit
the same sector and thus results obtained using sector information tend to be
quite conservative because they need to hold for _all_ nonlinearities in the
same sector. To overcome this, various researchers ([22, 44, 24]) have used
the concept of _slope restrictions_ to lessen the conservatism of the results.
A slope restricted nonlinearity is one where the generalised derivative of the
nonlinearity is assumed to lie between certain bounds. Denoting
$\partial\phi(.)$ as the generalised derivative of $\phi(.)$, the slope
restriction is equivalent to requiring

$\partial\phi\in[0,\alpha]$

which is often written as

$\displaystyle 0\leq\frac{\phi(x)-\phi(y)}{x-y}\leq\alpha\quad\forall
x,y\in\mathbb{R}$ (4)

where the lower bound on the slope has been taken to be zero (again possibly
by a loopshift). This therefore implies (setting $y=0$ and under the
assumption that $\phi(0)=0$) that $\phi\in{\rm Sector}[0,\alpha]$, but the
converse is not generally true.

Two final classes of nonlinearities are needed in order to arrive at the
Zames-Falb results: the classes of _monotone_ and _odd monotone_
nonlinearites. These classes of nonlinearities are essentially 2nd-4th
quadrant (odd) nonlinearities with a linear bound on their “gain”. They are of
independent interest, but are also the stepping stone to perhaps the most
useful Zames-Falb results: those devoted to slope-restricted nonlinearities.
Formally, these classes of nonlinearities are defined below.

###### Definition 1

A nonlinearity $\phi(\cdot)$ is said to be a _monotone non-decreasing_
nonlinearity if

$\Big{(}\phi(x)-\phi(y)\Big{)}(x-y)\geq 0\quad\forall x,y\in\mathbb{R}$

and it is bounded $|\phi(x)|\leq\gamma|x|$ for some $\gamma>0$ and all
$x\in\mathcal{L}_{2}$. A nonlinearity satisfying these properties is said to
belong to $\mathcal{N}_{M}$. In addition if $\phi(.)$ is odd, that is
$\phi(x)=-\phi(-x)$, then the nonlinearity is said to belong to
$\mathcal{N}_{MO}$.

Although, strictly speaking, the above definition defines monotone _non-
decreasing_ nonlinearities, with some abuse of terminology, they are often
described as _monotone increasing_ nonlinearities: in this article no
distinction will be made.

###  2.3 A few further concepts and notation

A few further concepts need emphasizing or introducing. In addition to the
plant $\mathbf{G}$, it is often necessary to introduce other linear operators
which will be denoted in bold font. A linear operator $\mathbf{P}$ will have
an associated Laplace transform $P(s)$, Fourier transform $P(j\omega)$ and
impulse response $p(t)$. It is recalled that the output (in the time-domain)
of such a system subject to an input $u(t)$ may be calculated using the
convolution integral

$y(t)=\int_{-\infty}^{\infty}g(\tau)u(t-\tau)d\tau$

When $\mathbf{G}$ is causal and the output at time $t=0$ is zero, this can be
replaced by

$y(t)=\int_{0}^{t}g(\tau)u(t-\tau)d\tau$

The Fourier transform of a signal $u(t)$, denoted $\hat{u}(j\omega)$, is
needed for the IQC approach introduced shortly.

Stability analysis based on input-output notions of passivity is intrinsically
linked to the concept of an inner product and this is also used frequently in
the IQC approach. The inner product of two signals $g(t)$ and $h(t)$ is
defined as

$\langle g(t),h(t)\rangle=\int_{-\infty}^{\infty}g(t)h(t)~{}dt$

By Parseval’s Identity (e.g. Appendix B in [10]), this is equivalent to the
frequency domain inner product

$\langle\hat{g}(j\omega),\hat{h}(j\omega)\rangle=\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat{g}^{*}(j\omega)\hat{h}(j\omega)~{}d\omega$

where $(\cdot)^{*}$ represents complex conjugate transpose. Obviously if the
time-domain inner product is non-negative, then so is the frequency domain
inequality.

The main technical idea of the Zames-Falb approach is to ensure a certain
integral is non-negative. It transpires that this is the case if the so-called
$\mathcal{L}_{1}$ norm of a certain “system” (linear operator) is less than
unity. Given a linear operator $\mathbf{P}$, the $\mathcal{L}_{1}$ norm is
defined as

$\|\mathbf{P}\|_{1}=\int_{-\infty}^{\infty}|p(t)|dt$

In the subsequent sections it will be seen that this norm arises naturally in
the analysis.

###  2.4 Multipliers

Figure 3:  Lur’e system with multipliers: Objects $\mathbf{M}$, called
_multipliers_ , assist with stability assessment  ††margin:  _In those days
spirits were brave, the stakes were high, men were real men, women were real
women and small furry creatures from Alpha Centauri were real small furry
creatures from Alpha Centauri._

There are various approaches one might take to prove stability of the system
in Figure 1. In the 1960s and 1970s the favoured approach was to use
_multipliers_ , which is where the term _Zames-Falb multiplier_ originates.
The idea with multipliers effectively comes from the desire to overcome
limitations imposed by _passivity_. Referring again to Figure 1, it is well
known that the (negative) feedback interconnection is stable if $\mathbf{G}$
and $\phi(\cdot)$ are both passive: actually it is more subtle, $\mathbf{G}$
must be strictly passive, but a discussion of this is out of scope. The key
point to note is that passivity places a very strong requirement on
$\mathbf{G}$ which is often not, and in practice rarely (never?), satisfied.
To overcome this strict requirement, one can introduce a so-called multiplier
$\mathbf{M}$, whose inverse $\mathbf{M}^{-1}$ is well-defined, and re-draw
Figure 1 as in Figure 3. Using the same passivity arguments it follows that
the interconnection is stable if the LTI system
$\mathbf{\tilde{G}}=\mathbf{MG}$ is strictly passive and the new nonlinearity
$\tilde{\phi}=\phi\circ\mathbf{M}^{-1}$ is passive. The advantage of this new
arrangement is that by introducing the multiplier $\mathbf{M}$ the passivity
requirements are on $\mathbf{GM}$ and are hopefully easier to enforce.

Remark: informality. This section has been written in an informal manner.
These concepts are used extensively in classical texts describing Zames-Falb
multipliers [41, 10] as well as in the original papers on the subject.
However, the approach taken in this article differs somewhat to that described
above. $\square$

###  2.5 Integral quadratic constraints (IQC’s)

Instead of using multipliers directly, the proofs given here are based on the
IQC approach to stability analysis [21]. The IQC approach is powerful because
it a) captures existing results in robust control; and b) enables one to
combine different stability results into a common analysis framework: The
Zames-Falb multiplier results are a good fit to this. A further advantage of
the IQC results is that the approach is arguably more direct than the
multiplier approach and dispenses with some of the loopshifting involved.

The key to applying IQC results is to establish that a nonlinearity satisfies
an IQC. In the frequency domain the nonlinearity $\phi(\cdot)$ is said to
_satisfy the IQC defined by $\Pi(j\omega)$_, where
$\Pi(j\omega)=\Pi^{*}(j\omega)$ and ${\Pi}\in\mathcal{RL}_{\infty}$, if the
following inequality holds for all $\hat{v}(j\omega)$:

$\displaystyle\int_{-\infty}^{\infty}\begin{bmatrix}\hat{v}(j\omega)\\\
\widehat{\phi(v(j\omega))}\end{bmatrix}^{*}\Pi(j\omega)\begin{bmatrix}\hat{v}(j\omega)\\\
\widehat{\phi(v(j\omega))}\end{bmatrix}~{}d\omega\geq 0$ (5)

Such an IQC is often established, and will be in this article, by first
deriving a time-domain integral quadratic constraint and then converting this
into the frequency domain form (5) via Parseval’s Identity.

The beauty of establishing an IQC of the form (5) is that the IQC theory
established in [21] can then be directly applied in order to obtain a
frequency domain stability condition, as indicated in the theorem below.

######  Theorem 1 ([21])

Let $\mathbf{G}\in\mathcal{RH}_{\infty}$ and assume $\phi(\cdot)$ is a
bounded, causal operator. Assume that:

  * i)

The interconnection in Figure 1 with $\phi(\cdot)$ is well posed.

  * ii)

The IQC defined by $\phi(\cdot)$ is satisfied.

  * iii)

There exists an $\epsilon$ such that

$\displaystyle\begin{bmatrix}G(j\omega)\\\
I\end{bmatrix}^{*}\Pi(j\omega)\begin{bmatrix}G(j\omega)\\\
I\end{bmatrix}<-\epsilon I\quad\forall\omega\in\mathbb{R}$ (6)

Then the interconnection in Figure 1 is stable.

This is the main stability result which will be used in this article: it is a
considerable simplification of the original result in [21] and is allowed due
to the form of the nonlinearites $\phi(\cdot)$ and the IQC’s which will be
developed. In general, more attention needs to be paid to the well-posedness
of the feedback system and the form of the IQC - see [21, 16]. Much more could
be written about IQC’s but in an effort to keep the article reasonably brief
and readable, further discussion is omitted. An excellent reference on the
subject is [16] and interesting connections between time-domain and frequency
domain IQC’s are made in [30] (see also [25]). A recent review of IQC’s is
given in [39].

##  3 Monotone Nonlinearities

Zames and Falb developed analysis techniques for two important classes of
nonlinearity: monotone non-decreasing nonlinearities and slope restricted
nonlinearities. In addition, both odd and non-odd versions of these
nonlinearities were treated. However, Zames and Falb’s results are best
appreciated first for monotone nonlinearities, and these results can then be
adapted for slope-restricted nonlinearities. In this section it will be shown
how the monotone property can be used to establish an IQC for these classes of
nonlinearities.

###  3.1 A preliminary result involving convex functions

Firstly, note that if $\phi\in\mathcal{N}_{M}$, then the integral

$F(\sigma)=\int_{0}^{\sigma}\phi(x)dx$

is a _convex function_ , since its derivative exists and is monotone non-
decreasing by assumption, viz

$\frac{dF(\sigma)}{d\sigma}=\phi(\sigma)$

Convexity of $F(\sigma)$ implies that, for all
$\sigma_{1},\sigma_{2}\in\mathbb{R}$,

$\displaystyle(\sigma_{1}-\sigma_{2})F^{\prime}(\sigma_{1})\geq
F(\sigma_{1})-F(\sigma_{2})$ (7) $\displaystyle\Leftrightarrow$
$\displaystyle(\sigma_{1}-\sigma_{2})\phi(\sigma_{1})\geq
F(\sigma_{1})-F(\sigma_{2})$ (8)

where $F^{\prime}(\sigma_{1})$ represents the derivative of $F(\sigma)$ at
$\sigma_{1}$. Since this inequality holds for any
$\sigma_{1},\sigma_{2}\in\mathbb{R}$, then it holds for $\sigma_{1}=x(t)$ and
$\sigma_{2}=x(t-\tau)$, where $t$ and $\tau$ are arbitrary, which gives

$\displaystyle\big{(}x(t)-x(t-\tau)\big{)}\phi(x(t))\geq F(x(t))-F(x(t-\tau))$
(9)

Integrating both sides w.r.t. $t$ then gives ††margin:  _“If, he thought to
himself, an infinite improbability machine is a virtual impossibility, then it
must logically be a finite improbability.”_

$\displaystyle\int_{-\infty}^{\infty}\big{(}x(t)-x(t-\tau)\big{)}\phi(x(t))dt\geq\underbrace{\int_{-\infty}^{\infty}\Big{(}F(x(t))-F(x(t-\tau))\Big{)}dt}_{=0}$
(10) $\displaystyle\Rightarrow$
$\displaystyle\int_{-\infty}^{\infty}\big{(}x(t)-x(t-\tau)\big{)}\phi(x(t))dt\geq
0$ (11)

which can then be re-arranged to give the remarkable, and perhaps unexpected,
result that

$\displaystyle\int_{-\infty}^{\infty}x(t)\phi(x(t))dt\geq\int_{-\infty}^{\infty}x(t-\tau)\phi(x(t))dt$
(12)

for all $\tau$. Inequality (12) is sometimes called the _area inequality_.

In fact, it will also be useful to develop another version of the _area
inequality_ as well which applies in the special case that $\phi(\cdot)$ is an
odd function. First note that if $\phi(.)$ is odd then its integral,
$F(\cdot)$ is even. Hence replacing $\sigma_{2}$ with $-\sigma_{2}$ in
inequality (8) thus gives:

$\displaystyle(\sigma_{1}-(-\sigma_{2}))\phi(\sigma_{1})$ $\displaystyle\geq
F(\sigma_{1})-F(-\sigma_{2})$ (13)
$\displaystyle\Leftrightarrow(\sigma_{1}+\sigma_{2})\phi(\sigma_{1})$
$\displaystyle\geq F(\sigma_{1})-F(\sigma_{2})$ (14)

Again, noting that the above inequality holds for any
$\sigma_{1},\sigma_{2}\in\mathbb{R}$, then it holds for $\sigma_{1}=x(t)$ and
$\sigma_{2}=x(t-\tau)$, which gives

$\displaystyle\big{(}x(t)+x(t-\tau)\big{)}\phi(x(t))\geq F(x(t))-F(x(t-\tau))$
(15)

As before, integrating both sides w.r.t. $t$ then gives

$\displaystyle\int_{-\infty}^{\infty}\big{(}x(t)+x(t-\tau)\big{)}\phi(x(t))dt\geq\int_{-\infty}^{\infty}\Big{(}F(x(t))-F(x(t-\tau))\Big{)}dt$
(16) $\displaystyle\Rightarrow$
$\displaystyle\int_{-\infty}^{\infty}\big{(}x(t)+x(t-\tau))\phi(x(t))\big{)}dt\geq
0$ (17)

which can then be re-arranged to give a mirrored result to inequality (12):

$\displaystyle\int_{-\infty}^{\infty}x(t)\phi(x(t))dt\geq-\int_{-\infty}^{\infty}x(t-\tau)\phi(x(t))dt$
(18)

Combining inequalities (12) and (20) we have

$\displaystyle-\int_{-\infty}^{\infty}x(t)\phi(x(t))dt\leq\int_{-\infty}^{\infty}x(t-\tau)\phi(x(t))dt\leq\int_{-\infty}^{\infty}x(t)\phi(x(t))dt$
(19)

which of course implies that

$\displaystyle\left|\int_{-\infty}^{\infty}x(t-\tau)\phi(x(t))dt\right|\leq\int_{-\infty}^{\infty}x(t)\phi(x(t))dt$
(20)

which is an _area inequality_ for odd nonlinearities. The two area
inequalities can be packaged into the following Lemma.

######  Lemma 1 (Area Lemma)

Consider the nonlinearity $\phi(\cdot)\in\mathcal{N}_{M}$. Then, for all
$\tau\in\mathbb{R}$, inequality (12) holds. Furthermore, if
$\phi(\cdot)\in\mathcal{N}_{MO}$ then, for all $\tau\in\mathbb{R}$, inequality
(20) holds.

This result is covered in [44, 10, 41] and as Willems notes, has its roots in
re-arrangement inequalities [14].

###  3.2 Zames-Falb IQC for monotone nonlinearities

This section establishes an IQC involving a dynamic multiplier M, with
transfer function $M(s)=1-H(s)$ and impulse response $m(t)=\delta(t)-h(t)$,
for the nonlinearity $\phi\in\mathcal{N}_{M}$. Consider the integral

$\displaystyle I_{1}=$ $\displaystyle\langle\mathbf{M}x(t),\phi(x(t))\rangle$
(21) $\displaystyle=$
$\displaystyle\langle(I-\mathbf{H})x(t),\phi(x(t))\rangle$ (22)
$\displaystyle=$ $\displaystyle\langle
x(t),\phi(t)\rangle-\langle\mathbf{H}x(t),\phi(x(t))\rangle$ (23)
$\displaystyle=$ $\displaystyle\langle
x(t),\phi(t)\rangle-\int_{-\infty}^{\infty}\left(\int_{-\infty}^{\infty}h(\tau)x(t-\tau)d\tau\right)\phi(x(t))dt$
(24)

Changing the order of integration then gives

$\displaystyle I_{1}=$ $\displaystyle\langle
x(t),\phi(t)\rangle-\int_{-\infty}^{\infty}h(\tau)\left(\int_{-\infty}^{\infty}x(t-\tau)\phi(x(t))dt\right)d\tau$
(25)

Next, assuming $h(t)\geq 0\quad\forall t$, the area inequality (12) means that

$\displaystyle h(\tau)\int_{-\infty}^{\infty}x(t)\phi(x(t))dt\geq
h(\tau)\int_{-\infty}^{\infty}x(t-\tau)\phi(x(t))dt$ (26)

for all $\tau$, which then implies

$\displaystyle I_{1}$ $\displaystyle\geq\langle
x(t),\phi(t)\rangle-\int_{-\infty}^{\infty}h(\tau)\left(\int_{-\infty}^{\infty}x(t)\phi(x(t))dt\right)d\tau$
(27) $\displaystyle=\langle
x(t),\phi(t)\rangle-\int_{-\infty}^{\infty}h(\tau)d\tau\int_{-\infty}^{\infty}x(t)\phi(x(t))dt$
(28) $\displaystyle=\langle x(t),\phi(t)\rangle-\|\mathbf{H}\|_{1}\langle
x(t),\phi(x(t))\rangle$ (29) $\displaystyle=(1-\|\mathbf{H}\|_{1})\langle
x(t),\phi(t)\rangle$ (30)

Now since $\langle x(t),\phi(t)\rangle\geq 0$, we see that $I_{1}\geq 0$ if
$\|\mathbf{H}\|_{1}\leq 1$. The results can be summarised in the following
theorem.

######  Theorem 2 (Monotone Nonlinearities)

Let $\phi\in\mathcal{N}_{M}$, and let $\mathbf{M}=I-\mathbf{H}$ be such that
$h(t)\geq 0$ for all $t\in\mathbb{R}$ and $\|\mathbf{H}\|_{1}\leq 1$, then the
following time-domain integral quadratic constraint holds:

$\int_{-\infty}^{\infty}(\mathbf{M}x(t))\phi(x(t))\geq 0$

It is more usual to express this in the frequency domain and, indeed using
Parseval’s Identity, it follows that above “time-domain” IQC can be written in
the more familiar frequency domain form as

$\displaystyle\int_{-\infty}^{\infty}\begin{bmatrix}\hat{x}(j\omega)\\\
\widehat{\phi(x)}(j\omega)\end{bmatrix}^{*}\begin{bmatrix}0&M^{*}(j\omega)\\\
M(j\omega)&0\end{bmatrix}\begin{bmatrix}\hat{x}(j\omega)\\\
\widehat{\phi(x)}(j\omega)\end{bmatrix}d\omega\geq 0$ (31)

###  3.3 Zames-Falb IQC for monotone odd nonlinearities

††margin:  _“Listen, three eyes,” he said, “don’t you try to outweird me, I
get stranger things than you free with my breakfast cereal.”_

The strange thing about Theorem 2 is that there is a requirement for the
impulse response of the dynamic element of the multiplier (that is,
$\mathbf{H}$) to be non-negative: $h(t)\geq 0$ for all time. If more is known
about the nonlinearity, namely that it is odd and hence
$\phi\in\mathcal{N}_{MO}$, the requirement that $h(t)\geq 0$ can be omitted.
Consider again the integral:

$\displaystyle I_{1}=$ $\displaystyle\langle\mathbf{M}x(t),\phi(x(t))\rangle$
(32) $\displaystyle=\langle
x(t),\phi(t)\rangle-\int_{-\infty}^{\infty}h(\tau)\left(\int_{-\infty}^{\infty}x(t-\tau)\phi(x(t))dt\right)d\tau$
(33)

where the same steps have been applied as in the previous section. Previously,
at this point, positivity of $h(t)$ was invoked, but suppose this is not
assumed; instead use will be made of the other version of the area inequality
- inequality (20). From equation (33) it follows that

$\displaystyle I_{1}$ $\displaystyle\geq\langle
x(t),\phi(t)\rangle-\left|\int_{-\infty}^{\infty}h(\tau)\left(\int_{-\infty}^{\infty}x(t-\tau)\phi(x(t))dt\right)d\tau\right|$
(34) $\displaystyle\geq\langle
x(t),\phi(t)\rangle-\int_{-\infty}^{\infty}|h(\tau)|\left|\int_{-\infty}^{\infty}x(t-\tau)\phi(x(t))dt\right|d\tau$
(35)

Using the second version of the area inequality (20) yields

$\displaystyle I_{1}$ $\displaystyle\geq\langle
x(t),\phi(t)\rangle-\int_{-\infty}^{\infty}|h(\tau)|\left(\int_{-\infty}^{\infty}x(t)\phi(x(t))dt\right)d\tau$
(36) $\displaystyle=\langle x(t),\phi(t)\rangle-\|\mathbf{H}\|_{1}\langle
x(t),\phi(t)\rangle$ (37) $\displaystyle=(I-\|\mathbf{H}\|_{1})\langle
x(t),\phi(t)\rangle$ (38)

Again, note that if $\|\mathbf{H}\|\leq 1$ it is clear that $I_{1}\geq 0$.
This result can be stated as the following theorem:

######  Theorem 3 (Monotone Odd Nonlinearities)

Let $\phi\in\mathcal{N}_{MO}$, and let $\mathbf{M}=I-\mathbf{H}$ and be such
that $\|\mathbf{H}\|_{1}\leq 1$, then the following time-domain integral
quadratic constraint holds:

$\int_{-\infty}^{\infty}(\mathbf{M}x(t))\phi(x(t))\geq 0$

Again, this time-domain integral can be expressed as a more familiar frequency
domain IQC (31).

##  4 Slope-restricted nonlinearities

Most frequently, the Zames-Falb results are applied to systems where the
nonlinearity $\phi(\cdot)$ is a _slope-restricted_ nonlinearity: this means
that the generalised derivative of the nonlinearity lies between two values.
In this work, the lower slope of the nonlinearity is $0$ and the upper slope
is some positive number $\alpha$: $\partial\phi\in[0,\alpha]$. A little
generality is lost with this, but again, loopshifting can be used to guarantee
this; the exposition is also simplified with this assumption.

The trick to applying Zames-Falb multiplier results to systems with slope
restrictions is essentially one of re-arranging the system so that instead of
working with the original nonlinearity, a modified nonlinearity is created
such that it is simply monotone increasing: this then enables the original
results (on monotone nonlinearities) to be applied to the case of slope-
restricted nonlinearities. In the original results of Zames and Falb ([44] \-
see also [10]) the transformation of the original nonlinearity into a modified
nonlinearity was done using the commonly used absolute stability trick of
_loopshifting_. The same approach is effectively used here, but the results
are a little more direct.

###  4.1 Zames-Falb IQC for slope-restricted nonlinearities

Figure 4:  A loopshifted nonlinearity: $\phi(\cdot)$ is slope-restricted,
$\tilde{\phi}(\cdot)$ is monotone. The idea is to establish an IQC for
$\tilde{\phi}(\cdot)$ and then use this to find one for $\phi(\cdot)$.

Consider the modified nonlinearity shown in Figure 4. This nonlinearity,
$\tilde{\phi}$ can be written as

$\displaystyle\tilde{\phi}(\tilde{v})=\left\\{\begin{array}[]{ccl}\phi(\tilde{v})&=&\phi(v)\\\
v&=&\tilde{v}+\frac{1}{\alpha}\phi(v)\end{array}\right.$ (41)

Assuming solutions exist to this implicit equation, one can write:

$\displaystyle\tilde{\phi}(\tilde{v})=\phi\circ(1-\frac{1}{\alpha}\phi(.))^{-1}(\tilde{v})$
(42)

Note that because $\alpha\sigma-\phi(\sigma)$ is a monotone nonlinearity (by
virtue of the slope restriction $\partial\phi\in[0,\alpha]$), then

$1-\frac{1}{\alpha}\phi(.)$

also is and so is its inverse. Furthermore, again by the slope restriction,
$\phi(.)$ is monotonically increasing function, and hence so is $\tilde{\phi}$
defined above. From Section 3.2, on monotone nonlinearities, it follows that
$\tilde{\phi}$ therefore satisfies the IQC (31) where $\mathbf{H}$ should have
a positive impulse response. Note that this IQC essentially defines a
relationship between $\tilde{\phi}$ and $\tilde{v}$: one between $\phi$ and
$v$ is actually what is sought. However, from equation (41), the following
relationship holds:

$\displaystyle\begin{bmatrix}v\\\
\phi(v)\end{bmatrix}=\underbrace{\begin{bmatrix}1&\frac{1}{\alpha}\\\
0&1\end{bmatrix}}_{X}\begin{bmatrix}\tilde{v}\\\
\tilde{\phi}(\tilde{v})\end{bmatrix}$ (43)

Therefore if $\tilde{\phi}(.)$ satisfies the IQC defined by $\Pi(j\omega)$
given in equation (31), then the following inequality holds

$\displaystyle\int_{-\infty}^{\infty}\begin{bmatrix}\tilde{v}(j\omega)\\\
\tilde{\phi}(j\omega)\end{bmatrix}^{*}\begin{bmatrix}0&M^{*}(j\omega)\\\
M(j\omega)&0\end{bmatrix}\begin{bmatrix}\tilde{v}(j\omega)\\\
\tilde{\phi}(j\omega)\end{bmatrix}~{}d\omega\geq 0$ (44)

The relationship in equation (43) can be reversed to get

$\displaystyle\begin{bmatrix}\tilde{v}\\\
\tilde{\phi}(\tilde{v})\end{bmatrix}=\underbrace{\begin{bmatrix}1&-\frac{1}{\alpha}\\\
0&1\end{bmatrix}}_{X^{-1}}\begin{bmatrix}v\\\ \phi(v)\end{bmatrix}$ (45)

and then used in inequality (44) to get

$\displaystyle\int_{-\infty}^{\infty}\begin{bmatrix}{v}(j\omega)\\\
{\phi}(j\omega)\end{bmatrix}^{*}\begin{bmatrix}1&0\\\
-\frac{1}{\alpha}&1\end{bmatrix}\begin{bmatrix}0&M^{*}(j\omega)\\\
M(j\omega)&0\end{bmatrix}\begin{bmatrix}1&-\frac{1}{\alpha}\\\
0&1\end{bmatrix}\begin{bmatrix}{v}(j\omega)\\\
{\phi}(j\omega)\end{bmatrix}~{}d\omega\geq 0$ (46)

Therefore $\phi$ satisfies the relationship defined by the matrix

$\displaystyle{\Pi}_{S}(j\omega)$
$\displaystyle=(X^{-1})^{\prime}\Pi(j\omega)X^{-1}$ (47)
$\displaystyle=\begin{bmatrix}1&0\\\
-\frac{1}{\alpha}&1\end{bmatrix}\begin{bmatrix}0&M^{*}(j\omega)\\\
M(j\omega)&0\end{bmatrix}\begin{bmatrix}1&-\frac{1}{\alpha}\\\
0&1\end{bmatrix}$ (48) $\displaystyle=\begin{bmatrix}0&M^{*}(j\omega)\\\
M(j\omega)&-\frac{1}{\alpha}(M^{*}(j\omega)+M(j\omega))\end{bmatrix}$ (49)

or, multiplying through by $\alpha>0$, the more familiar form emerges:

$\displaystyle{\Pi}_{S}(j\omega)=\begin{bmatrix}0&\alpha M^{*}(j\omega)\\\
\alpha M(j\omega)&-M^{*}(j\omega)-M(j\omega)\end{bmatrix}$ (50)

As before, this result can be packaged into the theorem below.

######  Theorem 4 (Slope restricted nonlinearities)

Let $\phi(\cdot)$ be such that $\partial\phi\in[0,\alpha]$, and let
$\mathbf{M}=I-\mathbf{H}$ be such that $h(t)\geq 0$ for all $t$ and
$\|\mathbf{H}\|_{1}\leq 1$. Then $\phi(\cdot)$ satisfies the IQC defined by
${\Pi}_{S}(j\omega)$ in inequality (50).

###  4.2 Zames-Falb IQC for odd slope-restricted nonlinearities

The previous subsection effectively explained how a slope-restricted
nonlinearity could be transformed into a monotone nonlinearity and how the
results of Section 3.2, or more precisely Theorem 2, could be used to derive
an IQC for the original slope restricted nonlinearity. The same is done in
this section, but the additional assumption that $\phi(\cdot)$ is odd will be
imposed: this then allows the results of Section 3.3 to be used, and in
particular Theorem 3, to be harnessed to provide a similar result.

Firstly, again consider the modified nonlinearity depicted in Figure 4. This
can again be described by the implicit equation (41) or, more compactly by
equation (42). Note that if it is assumed that $\phi(.)$ is odd, then this
implies that $\tilde{\phi}(\cdot)$ is also odd because the composition and
inverse of odd functions is also odd. Therefore, it follows that
$\tilde{\phi}(\cdot)$ is both odd and monotone (from the discussion in the
previous section). With this in mind, from Theorem 3, $\tilde{\phi}(.)$
satisfies the IQC defined by (31). In exactly the same way as in the previous
subsection, using the relationship (43), it thus follows that $\phi(\cdot)$
satisfies the IQC defined by (50) _except_ without the stipulation that the
impulse response of the dynamic part of the multiplier, $\mathbf{H}$, is
positive. The conclusion of this section can be summarised in the following
theorem.

######  Theorem 5 (Odd Slope restricted nonlinearities)

Let $\partial\phi\in[0,\alpha]$ be an odd nonlinearity, and let
$\mathbf{M}=I-\mathbf{H}$ with $\|\mathbf{H}\|_{1}\leq 1$. Then $\phi(.)$
satisfies the IQC defined by ${\Pi}_{S}(j\omega)$ in inequality (50).

###  4.3 Comparison of Results

Figure 5:  Venn diagram showing relationship between the various Zames-Falb
results, assuming the lower bound on the slope is zero.

The relationship between the results is perhaps best summarised by the Venn
diagram in Figure 5, assuming that the lower slope restriction is zero. Note
that Theorem 3 is a special case of Theorem 2 with the added restriction that
$\phi(\cdot)$ should be odd, but with the corresponding relaxation that the
impulse response of $\mathbf{H}$ need not be positive. Likewise Theorem 4 is a
special case of Theorem 3 but with the added restriction that $\phi(\cdot)$
should be odd, but again with the relaxation that the impulse response of
$\mathbf{H}$ need not be positive.

##  5 Notes and Further Reading

The purpose of these notes was to give a clear exposition of the theory behind
Zames-Falb multipliers. The subject is still an active area of research and
for more up-to-date results, one is advised to consult the latest literature
in this area. In this section a few brief additional comments are given but
the comments are indeed brief and are by no means self-contained.

###  5.1 Applying Zames-Falb multipliers - searches

††margin:  _Infinity is just so big that by comparison, bigness itself looks
really titchy. Gigantic multiplied by colossal multiplied by staggeringly huge
is the sort of concept we’re trying to get across here._

Using Zames-Falb multipliers for stability analysis is significantly more
difficult than using the Circle or Popov criteria because one has to somehow
_find or choose_ the dynamics of the Zames-Falb multiplier, $\mathbf{M}$. This
is made more difficult by the observation that they may be drawn from a _very
wide range of linear operators_ : it was noted in the original work of Zames
and Falb that these operators do not have to even be rational or causal
222Irrationality has been exploited in the work of [12, 7, 4].. The natural
question is then: how does one pick “the best” Zames-Falb multiplier for
stability analysis? Or more precisely: what is the Zames-Falb multiplier
associated with the largest slope restriction or the smallest
$\mathcal{L}_{2}~{}$gain (of an appropriate mapping)?

In the frequency domain, for slope-restricted nonlinearities, the IQC results
of Theorem 1 mean that the following inequality must be verified for some
$\epsilon>0$:

$\displaystyle\begin{bmatrix}G(j\omega)\\\
I\end{bmatrix}^{*}\begin{bmatrix}0&\alpha M(j\omega)^{*}\\\ \alpha
M(j\omega)&-M(j\omega)-M^{*}(j\omega)\end{bmatrix}\begin{bmatrix}G(j\omega)\\\
I\end{bmatrix}\leq-\epsilon$ (51) $\displaystyle M(j\omega)^{*}(\alpha
G(j\omega)^{*}-I)+(\alpha G(j\omega)-I)M(j\omega)\leq-\epsilon$ (52)

To assist in the search, the above inequality is often transformed into a
search over symmetric matrices using the KYP Lemma (see [26]). This yields a
matrix inequality of the form

$\displaystyle\begin{bmatrix}A^{\prime}\mathbf{P}+\mathbf{P}A&\mathbf{P}B+C^{\prime}\\\
B^{\prime}\mathbf{P}+C&D+D^{\prime}\end{bmatrix}<0\quad\mathbf{P}=\mathbf{P}^{\prime}$
(53)

Unfortunately, matrix inequality is bilinear because the state-space matrices
$(A,B,C,D)$ are affine functions of the multiplier parameters. Therefore, one
still needs to find some way in to search, efficiently, for an appropriate
multiplier. It is important to emphasize that:

_The frequency domain “passivity” constraint (52) must also be combined with a
time-domain constraint enforcing $\|\mathbf{H}\|_{1}\leq 1$ (sometimes called
the $\mathcal{L}_{1}$ condition)._

This observation is, effectively, what makes analysis using Zames-Falb
multipliers challenging in practice.

There are three prevailing approaches to Zames-Falb multiplier searches in the
literature. Perhaps the most common was proposed by [8] which, in essence,
requires one to choose multipliers of a particular form:

$\displaystyle M(s)=1-H(s)=1-\prod_{i=1}^{n}\frac{k_{i}}{s+a_{i}}$ (54)

Note that in the case that $\mathbf{M}\in\mathcal{RH}_{\infty}$ $a_{i}>0$ and
therefore choosing $k_{i}>0$ one immediately has that $h(t)\geq 0$ for all
$t$. The $\mathcal{L}_{1}$ condition then simply becomes a case of ensuring
$\|\mathbf{H}\|_{1}=\prod_{i=1}^{n}k_{i}/a_{i}\leq 1$. Equipped with this
form, one can then simply insert this multiplier in equation (50) and use this
to evaluate the frequency domain inequality in Theorem 1. Note that this
frequency domain inequality is equivalent to a matrix inequality by the KYP
Lemma [26]. The main issue with this approach is that one needs to choose the
poles of the multiplier $a_{i}$ to ensure that the arising matrix inequalities
are linear. The key advantage is that one has an exact expression for the
$\mathcal{L}_{1}$ norm of $\mathbf{H}$. References [40, 39] discuss this
approach further.

††margin:  _…man had always assumed that he was more intelligent than dolphins
because he had achieved so much - the wheel, New York, wars and so on - whilst
all the dolphins had ever done was muck about in the water having a good time.
But conversely, the dolphins had always believed that they were far more
intelligent than man - for precisely the same reasons._

An alternative approach to the above, which was originally suggested in [36]
and then used in [5, 4, 32, 33] and elsewhere, is to assume a particular order
of multiplier (at least plant-order) and then to use a change of variables
similar to that used in [29] to obtain convex conditions for stability. The
approach is algebraically involved and either uses a bound on the
$\mathcal{L}_{1}$ norm [29], which is known to be extremely conservative in
some cases; or one can use symmetric multipliers [32] which remove the
conservatism in the $\mathcal{L}_{1}$ norm, but then one must accept the phase
restrictions imposed by the symmetry of the multiplier. My own - and somewhat
biased view - is that this general approach is somewhat more intellectually
satisfying than the previous one where effectively one makes, in practice,
some arbitrary choices for poles and order - but improvements need to be made
to overcome the current conservatism which plagues the approach in some cases.

The final approach that is worth mentioning is that in Zames and Falb’s
original results, the multiplier $\mathbf{M}$ may actually be chosen
_irrational_ and this has been exploited by several authors [12] and [7] in
order to obtain stability results. Recently [4] Carrasco, Heath and co-authors
have proposed discrete-time search for multipliers, based on FIR functions,
which when interpreted in continuous time provides irrational multipliers
which appear to be competitive with the state of the art. The potential
disadvantage with this technique is that it may not be easy to combine this
with synthesis techniques.

Approach | Order |  $\mathcal{L}_{1}$ bound | Comments
---|---|---|---
[8, 11] | chosen | exact | User must choose order and poles
[36, 5] | plant order | upper bound | Line search required
[32] |  2$\times$ plant order | exact | Non-causal multipliers; symmetric state-space structure
[7] | irrational | exact | Competitive but perhaps fragile
Table 1:  Comparison between some different approaches for proving stability
with Zames-Falb multipliers

Table 1 gives a rough comparison between the three different approaches: a
detailed comparison is out of scope and the interested reader should consult
[6] or [11] for more detailed discussion. It is interesting that opinions on
the superiority of a particular form of multiplier vary and that the same
property may be perceived to be an advantage by some researchers, but a
disadvantage by others. At this point it is fair to say “the jury is out” on
which approach is best.

###  5.2 Multivariable systems

This article, and the original results of Zames and Falb, has been dedicated
to the case when $\phi(\cdot)$ is a scalar nonlinearity. More generally,
$\phi(\cdot):\mathbb{R}^{m}\mapsto\mathbb{R}^{m}$ and hence multivariable
generalisations of Zames-Falb multipliers must be used. This extension was
only carried out relatively recently by Safonov and co-authors in a number of
papers [28, 19, 20]. The generalisations are not trivial and require further
assumptions on the form of the nonlinearity to be made, but they do have IQC
interpretations in much the same way as their scalar-valued counterparts [9].
The search for multipliers is somewhat more involved however: the
generalisation of [36] to the multivariable case [38] is effectively too
conservative and too computationally demanding333The word “tractable” in the
title is, in hindsight, misleading; “intractable” might be more appropriate..
The approaches given in [11, 31] both have merit but require choices to be
made which are not entirely systematic. Therefore, while it seems that much of
the spirit of the Zames-Falb multipliers carries over to the multivariable
case, the search for suitable multipliers seems inherently more difficult
444This section needs improvement and more detail. Similar sentiments probably
apply to the article as a whole..

###  5.3 How conservative are Zames-Falb multipliers?

It has been known for some time that Zames-Falb multipliers are potentially
much less conservative than either the Circle or Popov criteria – in the sense
that they may conclude stability when the Circle/Popov Criteria fail to do so.
It is not yet certain whether a Zames-Falb Multiplier is a _necessary_ as well
as _sufficient_ condition for absolute stability. Results in both continuous
([18]) and discrete time ([45]) seem to suggest that the existence of a Zames-
Falb multiplier is fairly close to necessary555It is believed by some that
W.P. Heath’s dog has worked out the necessary conditions. for absolute
stability, and thus using Zames-Falb multipliers should give the control
engineer some confidence in the results obtained.

###  5.4 A historical note

An often overlooked contributor to the development of Zames-Falb multipliers
is the person at their origin: R O’Shea 666It was J. Carrasco and W.P. Heath
who impressed upon me the importance of R. O’Shea in the development of Zames-
Falb multipliers. His contribution is acknowledged in the paper [6].. In a few
papers [22, 23], O’Shea introduces the idea of the sort of multipliers which
have now become known as “Zames-Falb” multipliers. Although O’Shea did not
quite work out the precise mathematics of the multipliers, he should at least
be credited with their inception.

##  6 Conclusion

††margin:  _You live and learn. At any rate, you live._

The purpose of this article was to introduce the main technical ideas behind
Zames-Falb multiplier’s in a reasonably accessible way. After finalising the
article, it was evident that I skimmed over some topics and, in places,
introduced notation which was either excessive, unclear or both. These two
deficiencies have undoubtedly jeopardised my original intention of
readability. Then, of course, there are the inevitable typos which will have
exacerbated this. This document is, however, something of a working document
which I intend to update; in the next version I hope to address some of the
weaknesses of this version.

##  7 Acknowledgements

I have been studying Zames-Falb multipliers for over 10 years and during that
time I have greatly benefitted from discussions on the subject with Drs. J.
Carrasco, R. Drummond, J.M. Gomes da Silva Jr., W.P. Heath, M.L. Kerr, and J.
Sofrony, and also some of the students who took my Nonlinear Control course
and other colleagues at various conferences. I’d like to also thank Dr. A.L.J.
Bertolin for pointing out some glaring inconsitencies in Section 4.1 of a
previous version which have now (hopefully) been corrected. Her work on
controller design using Zames-Falb multipliers is of independent interest [2].

## References

  * [1] D. Adams.  The hitch hiker’s guide to the galaxy: a trilogy in five parts.  Random House, 1995.
  * [2] A.L.J. Bertolin, R.C.L.F. Oliveira, G. Valmorbida, and P.L.D. Peres.  Dynamic output-feedback control of continuous-time Lur’e systems using Zames-Falb multipliers by means of an LMI-based algorithm.  IFAC-PapersOnLine, 55(25):109–114, 2022.
  * [3] L.J. Bridgeman and J.R. Forbes.  Conic-sector-based control to circumvent passivity violations.  International Journal of Control, 87(8):1467–1477, 2014.
  * [4] J. Carrasco, W. P. Heath, J. Zhang, N. S. Ahmad, and S. Wang.  Convex searches for discrete-time Zames-Falb multipliers.  IEEE Transactions on Automatic Control, 65(11):4538–4553, 2020\.
  * [5] J. Carrasco, M. Maya-Gonzalez, A. Lanzon, and W.P. Heath.  LMI searches for anticausal and noncausal rational Zames–Falb multipliers.  Systems and Control Letters, 70:17–22, 2014.
  * [6] J. Carrasco, M.C. Turner, and W.P. Heath.  Zames–Falb multipliers for absolute stability: From O’Shea’s contribution to convex searches.  European Journal of Control, 28:1–19, 2016.
  * [7] M. Chang, R. Mancera, and M. Safonov.  Computation of Zames-Falb multipliers revisited.  IEEE Transactions on Automatic Control, 57(4):1024–1029, 2011.
  * [8] X. Chen. and T. Wen.  Robustness analysis of LTI systems with structured incrementally sector bounded nonlinearities.  American Control Conference, pages 3883–3887, 1995.
  * [9] F.J. D’Amato, M.A. Rotea, A.V. Megretski, and U.T. Jonsson.  New results for analysis of systems with repeated nonlinearities.  Automatica, 37(6):739–747, 2001.
  * [10] C. A. Desoer and M. Vidyasagar.  Feedback systems: input-output properties.  SIAM, 2009.
  * [11] M. Fetzer and C.W. Scherer.  Full-block multipliers for repeated, slope-restricted scalar nonlinearities.  International Journal of Robust and Nonlinear Control, 27(17):3376–3411, 2017.
  * [12] P.B. Gapski and J.C. Geromel.  A convex approach to the absolute stability problem.  IEEE Transactions on Automatic Control, 39(9):1929–1932, 1994.
  * [13] M. Green and D.J.N. Limebeer.  Linear Robust Control.  Prentice-Hall, Inc., 1994.
  * [14] G.F. Hardy, J.E. Littlewood, and G. Pólya.  Inequalities.  Cambridge University Press, 1952.
  * [15] W. P. Heath and A. G. Wills.  Zames-Falb multipliers for quadratic programming.  In IEEE Conference on Decision and Control, pages 963–968, 2005\.
  * [16] U. Jönsson.  Lecture notes on integral quadratic constraints.  2001\.
  * [17] H.K. Khalil.  Nonlinear Systems.  Prentice Hall, New Jersey, 1996.
  * [18] S. Z. Khong and L. Su.  On the necessity and sufficiency of the Zames-Falb multipliers.  arXiv preprint arXiv:2009.13571, 2020.
  * [19] V. V. Kulkarni and M.G. Safonov.  All multipliers for repeated monotone nonlinearities.  IEEE Transactions on Automatic Control, 47(7):1209–1212, 2002.
  * [20] R. Mancera and M.G. Safonov.  All repeated multipliers for MIMO nonlinearities.  Systems and Control Letters, 54(4):389–397, 2005.
  * [21] A.V. Megretski and A. Rantzer.  System analysis via integral quadratic constraints.  IEEE Transactions on Automatic Control, 42(6):819–830, 1997.
  * [22] R. O’Shea.  A combined frequency-time domain stability criterion for autonomous continuous systems.  IEEE Transactions on Automatic Control, 11(3):477–484, 1966.
  * [23] R. O’Shea.  An improved frequency time domain stability criterion for autonomous continuous systems.  IEEE Transactions on Automatic Control, 12(6):725 – 731, 1967.
  * [24] P.G. Park.  Stability criteria of sector and slope restricted Lur’e systems.  IEEE Trans. Aut. Control, 47(2):308–313, 2002.
  * [25] H. Pfifer and P.J. Seiler.  Less conservative robustness analysis of linear parameter varying systems using integral quadratic constraints.  International Journal of Robust and Nonlinear Control, 26(16):3580–3594, 2016.
  * [26] A. Rantzer.  On the Kalman-Yakubovich-Popov lemma.  Systems and Control Letters, 28(1):7–10, 1996.
  * [27] M.G. Safonov and M. Athans.  A multiloop generalization of the circle criterion for stability margin analysis.  IEEE Transactions on Automatic Control, 26(2):415–422, 1981.
  * [28] M.G. Safonov and V.V. Kulkarni.  Zames Falb multipliers for MIMO nonlinearities.  International Journal of Robust and Nonlinear Control, 10(10):1025–1038, 2000.
  * [29] C. Scherer, P. Gahinet, and M. Chilali.  Multiobjective output feedback control via LMI optimisation.  IEEE Transactions on Automatic Control, 42(7):896–911, 1997.
  * [30] P.J. Seiler.  Stability analysis with dissipation inequalities and integral quadratic constraints.  IEEE Transactions on Automatic Control, 60(6):1704–1709, 2015.
  * [31] M. C. Turner and R. Drummond.  Analysis of MIMO Lurie systems with slope restricted nonlinearities using concepts of external positivity.  In IEEE Conference on Decision and Control, pages 163–168, 2019\.
  * [32] M. C. Turner and R. Drummond.  Analysis of systems with slope restricted nonlinearities using externally positive Zames-Falb multipliers.  IEEE Transactions on Automatic Control, 65(4):1660–1667, 2020.
  * [33] M. C. Turner and R. Drummond.  Discrete-time systems with slope restricted nonlinearities: Zames–Falb multiplier analysis using external positivity.  International Journal of Robust and Nonlinear Control, 31(6):2255–2273, 2021.
  * [34] M.C. Turner and J. R. Forbes.  Conic sector analysis using integral quadratic constraints.  International Journal of Robust and Nonlinear Control, 30(2):741–755, 2020.
  * [35] M.C. Turner and M.L. Kerr.  $\mathcal{L}_{2}~{}$ gain bounds for systems with sector bounded and slope-restricted nonlinearities.  International Journal of Robust and Nonlinear Control, 22(13):1505–1521, 2012.
  * [36] M.C. Turner, M.L. Kerr, and I. Postlethwaite.  On the existence of stable, causal multipliers for systems with slope-restricted nonlinearities.  IEEE Transactions on Automatic Control, 54(11):2697–2702, 2009\.
  * [37] M.C. Turner, M.L. Kerr, and I. Postlethwaite.  Authors reply to “Comments on ‘On the Existence of Stable, Causal Multipliers for Systems with Slope-Restricted Nonlinearities’”.  IEEE Transactions on Automatic Control, 57(9):2428–2430, 2012.
  * [38] M.C. Turner, M.L. Kerr, and J. Sofrony.  Tractable stability analysis for systems containing repeated scalar slope-restricted nonlinearities.  International Journal of Robust and Nonlinear Control, 25(7):971–986, 2015.
  * [39] J. Veenman, C. W. Scherer, and H. Köroğlu.  Robust stability and performance analysis based on integral quadratic constraints.  European Journal of Control, 31:1–32, 2016.
  * [40] J. Veenman and C.W. Scherer.  IQC-synthesis with general dynamic multipliers.  International Journal of Robust and Nonlinear Control, 24(17):3027–3056, 2014.
  * [41] J. C. Willems.  The analysis of feedback systems.  MIT Press, 1971.
  * [42] G. Zames.  On the input-output stability of time-varying nonlinear feedback systems–part I: Conditions derived using concepts of loop gain, conicity, and positivity.  IEEE Transactions on Automatic Control, 11(2):228–238, 1966.
  * [43] G. Zames.  On the input-output stability of time-varying nonlinear feedback systems–part II: Conditions involving circles in the frequency plane and sector nonlinearities.  IEEE Transactions on Automatic Control, 11(3):465–476, 1966.
  * [44] G. Zames and P.L. Falb.  Stability conditions for systems with monotone and slope restricted nonlinearities.  SIAM Journal of Control, 6(1):89–108, 1968.
  * [45] J. Zhang, J. Carrasco, and W.P. Heath.  Duality bounds for discrete-time Zames-Falb multiplier.  arXiv preprint arXiv:2008.11975, 2020.
  * [46] K. Zhou, J.C. Doyle, and K. Glover.  Robust and Optimal Control.  Prentice Hall, 1996.