# On the characterization of butterfly and multi-loop hysteresis behavior

M.A. Vasquez-Beltran, B. Jayawardhana, R. Peletier *This paper is based on
research developed in the DSSC Doctoral Training Programme, co-funded through
a Marie Skłodowska-Curie COFUND (DSSC 754315).1M.A. Vasquez Beltran and B.
Jayawardhana are with the Engineering and Technology Institute Groningen,
Faculty of Science and Engineering, University of Groningen, 9747AG Groningen,
The Netherlands<EMAIL_ADDRESS>Peletier is
with the Kapteyn Astronomical Institute, Faculty of Science and Engineering,
University of Groningen, 9747AG Groningen, The Netherlands<EMAIL_ADDRESS>
###### Abstract

While it is widely used to represent hysteresis phenomena with unidirectional-
oriented loops, we study in this paper the use of Preisach operator for
describing hysteresis behavior with multidirectional-oriented loops. This
complex hysteresis behavior is commonly found in advanced materials, such as,
shape-memory alloys or piezoelectric materials, that are used for high-
precision sensor and actuator systems. We provide characterization of the
Preisach operators exhibiting such input-output behaviors and we show the
richness of the operators that are capable of producing intricate loops.

##  I Introduction

The term “hysteresis” comes from the Greek “to lag behind” and was originally
coined by Ewing in 1885 to describe a phenomenon occurring in the
magnetization process of soft iron caused by reversal and cyclic changes of
the input magnetic field. Currently, hysteresis is known to be present in
several classes of physical systems such as ferroelectric and ferromagnetic
materials, shape memory alloys, and mechanical systems with friction.
Hysteresis represents a quasi-static dependence between the input and output
of a system whose phase plot describes particular curves known as hysteresis
loops [1].

Hysteresis is a non-linear phenomenon which has been represented in various
different mathematical formulation. One major distinction in representing
hysteresis is between the physics-based models and phenomenological models.
The former focuses on describing the hysteresis phenomenon from the particular
physical relations of the system under consideration, whereas the latter
focuses on the empirical description of the input-output behavior. Due to the
simplicity and ability to encapsulate many typical characteristics of
hysteresis behavior, the phenomenological models have been widely studied for
the past decades. Two of the phenomenological models that have been widely
used are the Preisach operator [2, 3], whose formulation incorporates other
operator-based models such as the Prandlt operator; and the Duhem model [4]
whose formulation incorporates other models based on non-smooth integro-
differential equations such as the Bouc-Wen model and the Dahl model.

In literature, there are numerous works that investigate the mathematical
properties of these phenomenological models [5, 6, 7]. Subsequently, these
characterization works are directly applicable for the stability analysis of
and the control design for systems that consist of sub-systems exhibiting
hysteresis behavior. For instance, the construction of an approximate inverse
model is pursued in [8] in order to stabilize a control system containing
hysteretic elements. In recent years, it has been shown that these
phenomenological hysteresis models can exhibit passivity/dissipativity
property which is a typical property of physical systems. The dissipativity of
Duhem model has been shown in [9, 10] while that of Preisach model is
presented in [11]. These dissipativity properties are closely related to the
orientation of the hysteresis loops.

Despite these numerous endeavors, most of the works in hysteresis modeling
have focused on the characterization of the hysteresis behavior whose phase
plot describes single-oriented loop as illustrated in Fig. 1. Common examples
of single-oriented loop occurs in the relation between polarization and
electric fields of piezoelectric materials or the relation between
magnetization and magnetic field of magnetostrictive materials. However, there
exists another class of hysteresis behavior reported in literature (see, for
instance, [12, 13, 14, 15]) whose phase plot describes two loops with opposite
orientations and connected at an intersection point as depicted in Fig. 2.
From the resemblance to the wings of a butterfly, this behavior is known as
butterfly hysteresis behavior. Examples of this type of hysteresis behavior
occur in the relation between strain and electric field of piezoelectric
materials and the relation between strain and magnetic field of
magnetostrictive materials.

To the authors’ best knowledge, there are two works providing mathematical
analysis for the modeling of butterfly hysteresis behavior. Firstly, in [16] a
modified Bouc–Wen is studied which can describe a particular class of
asymmetric double hysteresis loop behavior by introducing position and/or
acceleration information into the model equation. Moreover, when the
parameters of the model satisfy particular conditions, it has been shown that
passivity property holds [17]. Secondly, in [18], a framework to transform
butterfly loops to single-oriented loops is proposed. Although this approach
facilitates the use of the well-studied hysteresis models and enables the
possibility of implementing some of its known control strategies in systems
including elements that exhibit hysteresis with butterfly loops, it relies on
the existence of a convex mapping and restricts the loop shape to have exactly
two minima with the same value.

In this work, we extend the results and include the proof of propositions in
[19] where we introduced a Preisach hysteresis operator capable of exhibiting
butterfly loops. It is used to model the relation between strain and electric
field of a particular piezoelectric material. We firstly present the analysis
of a class of Preisach operators whose weighting function has one positive and
one negative domain. We show that under mild assumptions over the
distributions of these domains, the input-output behavior of Preisach operator
can exhibits butterfly loops. Subsequently, we introduce a general class of
Preisach operator whose weighting function can assume more than one positive
and one negative domain. We show that the input-output behavior of these
operators can exhibit hysteresis loops with two or more sub-loops. Finally, we
present the stability analysis of a Lur’e system with multi-loop hysteresis
element based on the results in [20].

Figure 1: An illustration of a simple hysteresis loop $\mathcal{H}_{u,y}$ that
typically describes the relation between polarization and electric fields of
piezoelectric materials or the relation between magnetization and magnetic
field of magnetostrictive materials.  Figure 2: An illustration of a butterfly
loop $\mathcal{H}_{u,y}$ that can describe the relation between strain and
electric field of piezoelectric materials or the relation between strain and
magnetic field in magnetostrictive materials.

This paper is organized as follows. In Section II we give some preliminaries
that include the definition of hysteresis operator, operators with clockwise
and counterclockwise input-output behavior, and the standard definition of the
Preisach operator. In Section III, we present the Preisach butterfly operator
and the proof of the main results in [19]. In Section IV we introduce a
characterization of the self-intersections in a hysteresis loop and their
relation to the weighting function of a Preisach multi-loop operator and
Section V presents the absolute stability analysis of the Lur’e system with a
Preisach multi-loop operator. Finally, in Section VI we give the conclusions.

##  II Preliminaries

Notation. We denote the spaces of piecewise continuous, absolute continuous
and continuous differentiable functions $f:U\to Y$ by $C_{\text{pw}}(U,Y)$,
$AC(U,Y)$ and $C(U,Y)$, respectively. A function $f:U\to Y$ is monotonically
increasing (resp. decreasing) if for every $u_{1},u_{2}\in U$ such that
$u_{1}<u_{2}$ we have that $f(u_{1})\leq f(u_{2})$ (resp. $f(u_{1})\geq
f(u_{2})$).

###  II-A Clockwise, counterclockwise and butterfly loops

In order to define hysteresis operators (following the formulation in [21]),
we introduce below three auxiliary concepts: time-transformation, rate-
independent operator and causal operator.

###### Definition II.1

A function $\phi:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is called a time
transformation if $\phi(t)$ is continuous and increasing with $\phi(0)=0$ and
$\lim_{t\to\infty}\phi(t)=\infty$. $\hfill\triangle$

###### Definition II.2

An operator $\Phi$ is said to be rate independent if

$\big{(}\Phi(u\circ\phi)\big{)}(t)=\Phi(u)\circ\phi(t)$

holds for all $u\in AC(\mathbb{R}_{+},\mathbb{R})$, $t\in\mathbb{R}_{+}$ and
all admissible time transformation $\phi$. $\hfill\triangle$

###### Definition II.3

The operator $\Phi$ is said to be causal if for all $\tau>0$ and all
$u_{1},\,u_{2}\in AC(\mathbb{R}_{+},\mathbb{R})$ it holds that

$u_{1}(t)=u_{2}(t)\ \ \ \forall t\in[0,\tau]\\\
\Rightarrow\big{(}\Phi(u_{1})\big{)}(t)=\big{(}\Phi(u_{2})\big{)}(t)\ \ \
\forall t\in[0,\tau].$

$\hfill\triangle$

Based on the definitions above, a hysteresis operator is defined formally as
follows.

###### Definition II.4

An operator $\Phi$ is called a hysteresis operator if $\Phi$ is causal and
rate-independent. $\hfill\triangle$

For the past decades, hysteresis operators have been widely studied and
characterized (see, for instance, the exposition in [7, 22, 4]). When a
periodic input is applied to a hysteresis operator, the input-output phase
plot will undergo a periodic closed orbit which is commonly referred to as
hysteresis loop. Similar to the periodic input-output map introduced in [23,
Definition 2.2] for Duhem models, we can define a hysteresis loop as follows.

###### Definition II.5

Consider a hysteresis operator $\Phi$ and an input-output pair $(u,y)$ with
$y=\Phi(u)$. Let $u$ be periodic with a period of $T>0$, with one maximum
$u_{\max}\in\mathbb{R}$ and with one minimum $u_{\min}\in\mathbb{R}$ in its
periodic interval. Assume that there exists a constant $t_{p}\geq 0$ such that
$y$ is periodic in the interval $[t_{p},\infty)$. The periodic orbit given by
$\mathcal{H}_{u,y}=\left\\{(u(t),y(t))\ |\ t\in[t_{p},\infty)\right\\}$ is
called a hysteresis loop if there exists a $\upsilon\in\mathbb{R}$ such that

$\text{card}\big{(}\\{(\upsilon,\gamma)\in\mathcal{H}_{u,y}\ |\
\gamma\in\mathbb{R}\\}\big{)}=2,$

where card denotes the cardinality of a set. $\hfill\triangle$

In other words, the hysteresis loop as defined above means that the curve
defined by $\mathcal{H}_{u,y}$ can have at most two elements
$(\upsilon,\gamma_{1})$ and $(\upsilon,\gamma_{2})$ for any admissible point
$\upsilon$. This definition admits also multiple input-output loops that will
be studied further in this paper.

As shown in [10, 9, 24, 25], the input-output behavior of hysteresis operators
can be classified by the type of hysteresis loops they produce. Simple
hysteresis loops can have a clockwise or counterclockwise orientation which is
given in terms of the signed-area enclosed by its phase plot. Following from
Green’s theorem, the signed-area enclosed by an input-output pair $(u,y)$ that
forms a closed curve in an interval $[t_{1},t_{2}]$ is given by

$\mathbb{A}:=\frac{1}{2}\int_{t_{1}}^{t_{2}}\big{(}u(\tau)\dot{y}(\tau)-y(\tau)\dot{u}(\tau)\big{)}{\rm
d}\hbox{\hskip 0.5pt}\tau.$ (1)

Hence, generalizing this notion we can say that a hysteresis loop
$\mathcal{H}_{u,y}$ is clockwise (resp. counterclockwise) if its signed-area
$\mathbb{A}$ given by (1) with $t_{1}\geq t_{p}$ and $t_{2}=t_{1}+T$ satisfies
$\mathbb{A}<0$ (resp. $\mathbb{A}>0$). In a similar manner, we say that a
hysteresis operator $\Phi$ exhibits clockwise (resp. counterclockwise) input-
output behavior if there exists at least one hysteresis loop
$\mathcal{H}_{u,y}$ corresponding to an input-output pair $(u,y)$ with
$y=\Phi(u)$ which is clockwise (resp. counterclockwise).

Based on these concepts, we can study hysteresis operators $\Phi$ that give
rise to butterfly loops using the enclosed signed-area as in (1) of the
resulting hysteresis loops. However, as depicted in Fig. 2, we would like to
note that the so-called butterfly loops are composed of two sub-loops
connected by a self intersection point where one of loops is clockwise and the
other is counterclockwise. Thus, the total signed-area of the butterfly loop
could be either positive or negative depending on the difference between the
individual signed-area of each sub-loop. For this reason, we define a
butterfly hysteresis operator as follows.

###### Definition II.6

A hysteresis operator $\Phi$ is called a butterfly hysteresis operator if
there exists a hysteresis loop $\mathcal{H}_{u,y}$ with $y=\Phi(u)$ such that
$\mathbb{A}=0$, where $\mathbb{A}$ is defined as in (1) with $t_{1}\geq t_{p}$
and $t_{2}=t_{1}+T$. $\hfill\triangle$

We remark that this definition of butterfly hysteresis operator is equivalent
to the one that we have introduced in [19, Definition 2.2]. In the present
paper we have used the concept of hysteresis loop in Definition II.5 for
defining the butterfly hysteresis operator with a particular input-output pair
$(u,y)$ that forms a periodic orbit, e.g. on the interval $[t_{1},t_{1}+T]$.

###  II-B The clockwise and counterclockwise relay operator

One of the simplest hysteresis operators is the relay operator which we
introduce below in its counterclockwise and clockwise versions. We define the
counterclockwise relay operator
$\mathcal{R}^{\circlearrowleft}_{\alpha,\beta}:AC(\mathbb{R}_{+},\mathbb{R})\times\\{-1,1\\}\to
C_{\text{pw}}(\mathbb{R}_{+},\mathbb{R})$ with switching parameters
$\alpha>\beta$ and initial condition $r_{0}$ by

$\big{(}\mathcal{R}^{\circlearrowleft}_{\alpha,\beta}(u,r_{0})\big{)}(t):=\left\\{\begin{array}[]{ll}+1&\text{if
}u(t)>\alpha,\\\ -1&\text{if }u(t)<\beta,\\\
\big{(}\mathcal{R}^{\circlearrowleft}_{\alpha,\beta}(u,r_{0})\big{)}(t_{-})&\begin{aligned}
\text{if }&\beta\leq u(t)\leq\alpha,\\\ &\quad\text{and }t>0,\end{aligned}\\\
r_{0}&\begin{aligned} \text{if }&\beta\leq u(t)\leq\alpha,\\\ &\quad\text{and
}t=0.\end{aligned}\end{array}\right.$ (2)

Figure 3: Input-output phase plot of the counterclockwise relay operator
$\mathcal{R}^{\circlearrowleft}_{\alpha,\beta}$ as defined in (2)

Similarly, we define the clockwise relay operator
$\mathcal{R}^{\circlearrowright}_{\alpha,\beta}:AC(\mathbb{R}_{+},\mathbb{R})\times\\{-1,1\\}\to
C_{\text{pw}}(\mathbb{R}_{+},\mathbb{R})$ with switching parameters
$\alpha>\beta$ and initial condition $r_{0}$ by

$\big{(}\mathcal{R}^{\circlearrowright}_{\alpha,\beta}(u,r_{0})\big{)}(t):=\left\\{\begin{array}[]{ll}-1&\text{if
}u(t)>\alpha,\\\ +1&\text{if }u(t)<\beta,\\\
\big{(}\mathcal{R}^{\circlearrowright}_{\alpha,\beta}(u,r_{0})\big{)}(t_{-})&\begin{aligned}
\text{if }&\beta\leq u(t)\leq\alpha,\\\ &\quad\text{and }t>0,\end{aligned}\\\
-r_{0}&\begin{aligned} \text{if }&\beta\leq u(t)\leq\alpha,\\\ &\quad\text{and
}t=0.\end{aligned}\end{array}\right.$ (3)

We note that for a specified initial condition $r_{0}\in\\{-1,+1\\}$ both
relay operators are hysteresis operators in the sense of Definition II.4 with
the form

$\displaystyle\Phi(u)=\mathcal{R}_{\alpha,\beta}^{\circlearrowleft}(u,\,r_{0})\qquad\text{and}\qquad\Phi(u)=\mathcal{R}_{\alpha,\beta}^{\circlearrowright}(u,\,r_{0}).$

The input-output phase plot of both relay operators is illustrated in Figs. 3
and 4. It can be observed from definitions (2) and (3) that for equal initial
condition $r_{0}$, the output of a counterclockwise relay is equivalent to the
negative output of a clockwise operator for every input, i.e.
$\mathcal{R}^{\circlearrowleft}_{\alpha,\beta}=-\mathcal{R}^{\circlearrowright}_{\alpha,\beta}$.

Figure 4: Input-output phase plot of the clockwise relay operator
$\mathcal{R}^{\circlearrowright}_{\alpha,\beta}$ as defined in (3)

###  II-C Preisach Operator

The Preisach operator is, roughly speaking, the weighted integral of all
infinitesimal (counterclockwise) relay operators, also known as hysterons,
whose switching values satisfy $\alpha>\beta$. To provide a formal definition
of this, we need to introduce two concepts. Firstly, we denote by $P$ the
admissible plane of relay operators defined by
$P:=\\{(\alpha,\beta)\in\mathbb{R}^{2}\ |\ \alpha>\beta\\}$ and it is commonly
referred to as the Preisach plane. Secondly, we denote $\mathcal{I}$ the set
of interfaces where each interface $L\in\mathcal{I}$ is a monotonically
decreasing staircase curve parameterized in the form $L=\\{\sigma(\gamma)\in
P\ |\ \gamma\in\mathbb{R}_{+}\\}$ by a function $\sigma(\gamma)\in
C(\mathbb{R}_{+},P)$ which satisfies
$\lim_{\gamma\to\infty}\lVert\sigma(\gamma)\rVert=\infty$ and
$\sigma(0)=(\alpha,\alpha)$ for some $\alpha\in\mathbb{R}$. By monotonically
decreasing $L$ we mean that for every pair
$(\alpha_{1},\beta_{1}),(\alpha_{2},\beta_{2})\in L$ we have that
$\beta_{1}\leq\beta_{2}$ implies $\alpha_{1}\geq\alpha_{2}$. Based on these
concepts, the Preisach operator
$\mathcal{P}:AC(\mathbb{R}_{+},\mathbb{R})\times\mathcal{I}\to
AC(\mathbb{R}_{+},\mathbb{R})$ is formally expressed by

$\displaystyle\big{(}\mathcal{P}(u,L_{0})\big{)}(t):=$ (4)
$\displaystyle\quad\iint\displaylimits_{(\alpha,\beta)\in P}\mu(\alpha,\beta)\
\big{(}\mathcal{R}^{\circlearrowleft}_{\alpha,\beta}(u,r_{\alpha,\beta}(L_{0}))\big{)}(t)\
{\rm d}\hbox{\hskip 0.5pt}{\alpha}{\rm d}\hbox{\hskip 0.5pt}\beta$

where $\mu\in C_{\text{pw}}(P,\mathbb{R})$ is a weighting function,
$L_{0}\in\mathcal{I}$ is the initial interface and
$r_{\alpha,\beta}:\mathcal{I}\to\\{-1,+1\\}$ is an auxiliary function that
determines the initial condition of every relay
$\mathcal{R}^{\circlearrowleft}_{\alpha,\beta}$ according to its position
$(\alpha,\beta)\in P$ with respects to the initial interface $L_{0}$ and is
defined by

$\displaystyle r_{\alpha,\beta}(L_{0}):=$
$\displaystyle\quad\left\\{\begin{array}[]{ll}+1&\begin{aligned} \text{if
}L_{0}\cap\left\\{(\alpha_{1},\beta_{1})\in P\ |\
\alpha\leq\alpha_{1},\,\beta\leq\beta_{1}\right\\}\neq\emptyset,\end{aligned}\\\
-1&\text{otherwise}.\end{array}\right.$

In other words, the value of the function $r_{\alpha,\beta}$ will be $+1$ if
the point $(\alpha,\beta)\in P$ is above the initial interface $L_{0}$, and
will be $-1$ if the point $(\alpha,\beta)\in P$ is below the initial interface
$L_{0}$. It is important to note from (2) that the actual initial state of a
relay operator $\mathcal{R}^{\circlearrowleft}_{\alpha,\beta}$ is determined
by $r_{0}$ only when $\beta\leq u(0)\leq\alpha$. This can produce an
inconsistency between the values of the function $r_{\alpha,\beta}(L_{0})$ and
the actual initial state of relay operators
$\big{(}\mathcal{R}^{\circlearrowleft}_{\alpha,\beta}(u,r_{\alpha,\beta}(L_{0}))\big{)}(0)$
with $\alpha<u(0)$ or $\beta>u(0)$. Therefore, for well-posedness, we assume
in general that the initial interface $L_{0}$ in the Preisach operator (4)
satisfies $\big{(}u(0),u(0)\big{)}\in L_{0}$. As with the relay operator, the
Preisach operator is a hysteresis operator in the sense of Definition II.4 for
specified initial conditions $L_{0}\in\mathcal{I}$ in the form
$\Phi(u)=\mathcal{P}(u,L_{0})$.

As one of the most important hysteresis operators, the dynamic behavior and
geometric interpretation of the Preisach operator defined by (4) has been
studied well in literature. Fundamentally, it can be said that the output of
the Preisach operator is determined instantaneously with the variations of the
input as all the relays in $P$ react instantaneously and simultaneously to the
applied input $u$. For this reason, the initial interface $L_{0}$ evolves
continuously and at every time instance $t\geq 0$ there exists an interface
$L_{t}\in\mathcal{I}$ that divides the Preisach plane into two subdomains
$P^{+}_{t}$ and $P^{-}_{t}$ where all relays $\mathcal{R}_{\alpha,\beta}$ with
$({\alpha,\beta})\in P^{+}_{t}$ are in state $+1$ while all relays
$\mathcal{R}_{\alpha,\beta}$ with $({\alpha,\beta})\in P^{-}_{t}$ are in state
$-1$.

##  III Preisach Butterfly Operator

The classical definition of the Preisach operator [2, 3] considers that the
weighting function $\mu$ is positive with counterclockwise relays as in (2).
This assumption restricts all realizable hysteresis loops to be
counterclockwise (for any periodic input-output pairs). On the other hands,
when negative weighting function $\mu$ is assumed (with the same
counterclockwise relays) then the realizable hysteresis loops become clockwise
(see [26]). This is mainly due to the fact that
$\mathcal{R}^{\circlearrowleft}_{\alpha,\beta}=-\mathcal{R}^{\circlearrowright}_{\alpha,\beta}$,
in which case, a Preisach operator with negative weighting function $\mu$ and
counterclockwise relays is equivalent to a Preisach operator with a positive
weighting function $\mu$ and clockwise relays.

Inspired by the latter observation and considering that a butterfly hysteresis
operator must exhibit both clockwise and counterclockwise input-output
behavior, it is intuitive to think that a Preisach operator with both
clockwise and counterclockwise relays and a sign-definite weighting function,
or equivalently with only clockwise or only counterclockwise relays and whose
weighting function $\mu$ has positive and negative domains, can, under certain
conditions, exhibit a butterfly hysteresis operator. We formalize this idea
introducing a Preisach operator whose weighting function $\mu$ has particular
structure and whose input-output behavior exhibits a butterfly loop that
satisfies the zero signed-area condition of Definition II.6. For this purpose,
let us introduce the following lemma that allows us to compute the enclosed
area of the hysteresis loop of a single relay operator.

###### Lemma III.1

Consider the counterclockwise relay operator
$\mathcal{R}^{\circlearrowleft}_{\alpha,\beta}$ as in (2) with $\alpha>\beta$.
For every periodic signal $u$ with a period of $T$ and with one maximum
$u_{\max}\in\mathbb{R}$ and one minimum $u_{\min}\in\mathbb{R}$ in the
periodic interval, the signed-area $\mathbb{A}$ corresponding to the input-
output pair $(u,y)$ with
$y=\mathcal{R}^{\circlearrowright}_{\alpha,\beta}(u,r_{0})$ and
$r_{0}\in\\{-1,+1\\}$ is given by

$\mathbb{A}=\left\\{\begin{array}[]{ll}2(\alpha-\beta)&\text{if
}u_{\min}<\beta\text{ and }u_{\max}>\alpha\\\
0&\text{otherwise.}\end{array}\right.$

###### Proof:

It can be checked that the influence of relay initial condition to the output
signal $y=\mathcal{R}^{\circlearrowright}_{\alpha,\beta}(u,r_{0})$ disappears
after one period. In other words, the pair $(u,y)$ will form a hysteresis loop
$\mathcal{H}_{u,y}$ or a line in the time interval $[t_{1},\infty)$ with
$t_{1}\geq t_{p}=T$. When the range of the input covers both switching points
of the relay, i.e., $u_{\min}<\beta$ and $u_{\max}>\alpha$, then the relay
will switch periodically forming a hysteresis loop $\mathcal{H}_{u,y}$ as in
Fig. 3. Otherwise, it will be a line. If it is a hysteresis loop
$\mathcal{H}_{u,y}$ then the signed-area that is produced is given by the area
of the corresponding rectangle in the phase plot of $\\{(u(\tau),y(\tau))\ |\
\tau\in[t_{1},t_{1}+T]\\}$ which is equal to $2(\alpha-\beta)$. If it is a
line then the signed-area is equal to zero. ∎

An immediate consequence of Lemma III.1, is that the signed-area enclosed by
the hysteresis loop of a clockwise relay operator
$\mathcal{R}_{\alpha,\beta}^{\circlearrowright}$ defined by (3) is given by
$-2(\alpha-\beta)$ when $u_{\min}<\beta$ and $u_{\max}>\alpha$.

We recall now the following proposition from [19] which considers a particular
class of Preisach operator with two-sided weighting function $\mu$ whose
input-ouput behavior can exhibit butterfly loops. By two-sided we mean that
there exists a simple curve $B$ that divides the Preisach domain $P$ into two
disjoint subdomains $B_{+}$ and $B_{-}$ such that $P=B_{+}\cup B_{-}\cup B$
and where $\mu(\alpha,\beta)\geq 0$ for every $(\alpha,\beta)\in B_{+}$ and
$\mu(\alpha,\beta)\leq 0$ for every $(\alpha,\beta)\in B_{-}$.

###### Proposition III.2

([19, Proposition 3.1]) Consider a Preisach operator $\mathcal{P}$ as in (4)
with $\mu$ be a two-sided weighting function. Suppose that the first order
lower and upper partial moments of $\mu$ satisfy

$\displaystyle\int_{r}^{\infty}\mu(\alpha,\beta)\beta{\rm d}\hbox{\hskip
0.5pt}\beta$ $\displaystyle=\infty$ (5)
$\displaystyle\int_{-\infty}^{r}\mu(\alpha,\beta)\alpha{\rm d}\hbox{\hskip
0.5pt}\alpha$ $\displaystyle=\infty,$ (6)

for all $(\alpha,\beta)\in P$. Assume that the boundary curve $B$ is
monotonically decreasing. Then $\mathcal{P}$ is a butterfly hysteresis
operator.

###### Proof:

Let us take arbitrary $(\alpha_{1},\beta_{1})\in B$ with
$\alpha_{1}>\beta_{1}$ (i.e., the point $(\alpha_{1},\beta_{1})$ is not on the
boundary of $P$). Consider a subset of Preisach domain
$P_{1}:=\\{(\alpha,\beta)\in P\ |\ \alpha<\alpha_{1},\,\beta>\beta_{1}\\}$
which is a solid triangle whose vertices are at $(\alpha_{1},\beta_{1})$,
$(\alpha_{1},\alpha_{1})$ and $(\beta_{1},\beta_{1})$. Note that since $B$ is
monotonically decreasing, it separates $P$ in two polar regions where
weighting function $\mu$ assigned to the domain above $B$ has different sign
with that below $B$. Without loss of generality, we consider the case where
$B_{1-}$ is below $B$ and $B_{1+}$ is above $B$. The arguments below are still
valid when we consider the reverse case.

Due to the monotonicity of $B$, if we consider the extended area on left of
$P_{1}$, which is given by $P_{1-}^{\text{ext}}:=\\{(\alpha,\beta)\in P\ |\
\beta<\beta_{1},\alpha<\alpha_{1}\\}$, the weight $\mu$ in this area will have
the same sign as that in $B_{1-}$. The same holds for the extended area above
$P_{1}$ where the weight $\mu$ in $P_{1+}^{\text{ext}}:=\\{(\alpha,\beta)\in
P\ |\ \beta>\beta_{1},\alpha>\alpha_{1}\\}$ has the same sign as that in
$B_{1+}$.

Let us now analyze the input-output behavior when the input $u$ of the
Preisach operator is a periodic signal with a period of $T$ and with one
maximum $u_{\max}=\alpha_{1}$ and one minimum $u_{\min}=\beta_{1}$. It is
clear that for every $t\geq t_{p}=T$, the initial conditions of all relays in
$P_{1}$ no longer the affect output $y$ and it becomes periodic. Therefore,
the relays $\mathcal{R}^{\circlearrowleft}_{\alpha,\beta}$ whose states are
switching periodically correspond to the domain $P_{1}$ while the state of all
the relays in $P\backslash P_{1}$ remains the same as given by the initial
condition. Consequently, following from Lemma III.1, the signed-area of the
hysteresis loop $\mathcal{H}_{u,y}$ obtained from the input-output pair
$(u,y)$ is given by

$\displaystyle\mathbb{A}$
$\displaystyle=2\iint\displaylimits_{(\alpha,\beta)\in
P_{1}}\mu(\alpha,\beta)\ \big{(}\alpha-\beta\big{)}{\rm d}\hbox{\hskip
0.5pt}{\alpha}{\rm d}\hbox{\hskip 0.5pt}\beta$
$\displaystyle=2\underbrace{\iint\displaylimits_{(\alpha,\beta)\in
B_{1+}}|\mu(\alpha,\beta)|\big{(}\alpha-\beta\big{)}{\rm d}\hbox{\hskip
0.5pt}{\alpha}{\rm d}\hbox{\hskip 0.5pt}\beta}_{\mathbb{A}_{+}}$
$\displaystyle\qquad-2\underbrace{\iint\displaylimits_{(\alpha,\beta)\in
B_{1-}}|\mu(\alpha,\beta)|\big{(}\alpha-\beta\big{)}{\rm d}\hbox{\hskip
0.5pt}{\alpha}{\rm d}\hbox{\hskip 0.5pt}\beta}_{\mathbb{A}_{-}}$

When the variation of $\mu$ is such that $\mathbb{A}_{+}=\mathbb{A}_{-}$, we
have obtained the condition for $\mathcal{P}$ to be a butterfly hysteresis
operator where the chosen periodic input signal $u$ ensures that
$\mathbb{A}=0$. However since in general the variation in $\mu$ can be
asymmetric, the signed-area $\mathbb{A}_{+}$ may not be equal to
$\mathbb{A}_{-}$.

Let us consider the case when $\mathbb{A}_{-}>\mathbb{A}_{+}$ (i.e., the
negative weight is dominant in $P_{1}$). In this case, we modify the periodic
input signal $u$ such that its maximum $u_{\max}$ is parametrized by
$\lambda>\alpha_{1}$. Similar as before, we have that the relays
$\mathcal{R}^{\circlearrowleft}_{\alpha,\beta}$ whose states are switching
periodically correspond to the domains $P_{1}$ and
$P_{1+}^{\text{ext},\lambda}:=\\{(\alpha,\beta)\in P\ |\
\beta>\beta_{1},\lambda>\alpha>\alpha_{1}\\}$ while the state of relays
corresponding to $P\backslash(P_{1}\cup P_{1+}^{\text{ext},\lambda})$ remains
the same as given by the initial condition. Hence, using again Lemma III.1,
the signed-area of the hysteresis loop $\mathcal{H}_{u,y}$ corresponding to
the input-output pair $(u,y)$ with modified $u$ is now given by

$\displaystyle\mathbb{A}$
$\displaystyle=2(\mathbb{A}_{+}-\mathbb{A}_{-})+2\underbrace{\iint\displaylimits_{(\alpha,\beta)\in
P_{1+}^{\text{ext},\lambda}}|\mu(\alpha,\beta)|\big{(}\alpha-\beta\big{)}{\rm
d}\hbox{\hskip 0.5pt}{\alpha}{\rm d}\hbox{\hskip 0.5pt}\beta}_{h(\lambda)}.$

Since $\mu$ is a piecewise continuous function, the function $h(\lambda)$ is
also continuous function, $h(\alpha_{1})=0$ and is strictly increasing (as
$\mu>0$ in $P^{\text{ext}}_{1+}$). Due to the unboundedness of the first order
upper partial moment of $\mu$ as in (5), it follows that $h(\lambda)\to\infty$
as $\lambda\to\infty$. This implies that there exists $\alpha_{2}>\alpha_{1}$
such that $h(\alpha_{2})=\mathbb{A}_{-}-\mathbb{A}_{+}$. In this case, by
taking a periodic signal with its maximum $u_{\max}=\alpha_{2}$ and its
minimum $u_{\min}=\alpha_{1}$, the signed-area of the corresponding hysteresis
loop $\mathcal{H}_{u,y}$ is equal to zero as claimed.

On the other hand, when $\mathbb{A}_{+}>\mathbb{A}_{-}$, we can use vis-a-vis
similar arguments as above where $u_{\min}$ is now parametrized by
$\lambda<\beta_{1}$, instead of parameterizing $u_{\max}$ as before. For this
situation, the additional relays that are affected by the modified input
correspond to the domain $P_{1-}^{\text{ext},\lambda}:=\\{(\alpha,\beta)\in P\
|\ \lambda<\beta<\beta_{1},\alpha<\alpha_{1}\\}$. The claim then follows
similarly as above where the additional signed-area of the corresponding
hysteresis loop $\mathcal{H}_{u,y}$ is a continuous function $h(\lambda)$ that
is strictly increasing and approaches $-\infty$ as $\lambda\to-\infty$.

Finally, we can follow the same reasoning as above for the case when $B_{1-}$
is above $B$ and $B_{1+}$ is below $B$. ∎

In Proposition III.2, we consider a general case where $\mu$ can be any two-
sided function, as long as, its decay to zero, which is measured by its upper
and lower partial moments, is not too fast. If this condition is not
satisfied, we may not be able to find an extended subset in $P$ parametrized
by $\lambda$ (as used in the proof of Proposition III.2) such that the total
signed-area $\mathbb{A}$ of the hysteresis loop $\mathcal{H}_{u,y}$ with the
modified $u$ is zero. Nevertheless, the conditions over the upper and lower
partial moments of $\mu$ can be relaxed if we focus on a small region close to
the meeting point of $B$ and the line $\\{(\alpha,\beta)\ |\ \alpha=\beta\\}$.
This is the case for the class Preisach operator considered in the next
proposition which was also introduced in [19].

###### Proposition III.3

([19, Proposition 3.2]) Consider a Preisach operator $\Phi$ as in (4) with a
two-sided weighting function $\mu$. Assume that the boundary curve of $\mu$ is
given by $B=\\{(\alpha,\beta)\in P\ |\ \alpha=-\beta+\kappa\\}$ where
$\kappa\in\mathbb{R}_{+}$ is an offset and $\mu$ is anti-symmetric with
respect to $B$, i.e., $\mu(\alpha,\beta)=-\mu(-\alpha,-\beta)$ holds for all
$(\alpha,\beta)\in P$. Then $\Phi$ is a butterfly hysteresis operator.

###### Proof:

Proposition III.3 The proof of the proposition follows similarly to the proof
of Proposition III.2. In this case, it suffices to have a periodic input
signal $u$ whose maximum and minimum satisfy $u_{\max}=-u_{\min}+2\kappa$. The
relays $\mathcal{R}_{\alpha,\beta}$ whose state switches periodically will lie
in a subset of $P$ which has the form of isosceles and right triangle. Since
the weighting function is anti-symmetric with respect to $B$ then the signed-
area of the hysteresis loop $\mathcal{H}_{u,y}$ will be zero. ∎

As a particular case of study, we analyze in the next example a class of
Preisach butterfly operator in Proposition III.3 with symmetrical two-sided
weighting function.

###### Example III.4

Let $B:=\\{(\alpha,\beta)\in P\ |\ \alpha=-\beta\\}$ (with $\kappa=0$) and
consider a point $(-\beta_{1},\beta_{1})\in B$ such that
$P_{1}:=\\{(\alpha,\beta)\in P\ |\ \alpha<\beta_{1},\beta>-\beta_{1}\\}$. In
this case the subdomain of interest $P_{1}$ in the Preisach plane is an
isosceles triangle with vertices in $(\alpha_{1},\alpha_{1})$,
$(\alpha_{1},-\alpha_{1})$, and $(-\alpha_{1},-\alpha_{1})$. Let us define the
weighting function by

$\mu(\alpha,\beta):=\left\\{\begin{array}[]{ll}-1&\text{if }\alpha\leq-\beta,\
(\alpha,\beta)\in P_{1}\\\ 1&\text{if }\alpha>-\beta,\ (\alpha,\beta)\in
P_{1}\\\ 0&\text{otherwise }\end{array}\right.$ (7)

An illustration of the weighting function (7) is included in Fig. 5.

Figure 5: Weighting function $\mu(\alpha,\beta)$ defined by (7).

The Preisach operator with this weighting function clearly satisfies the
conditions of Proposition III.2 and consequently it is a butterfly hysteresis
operator. It follows from this proposition that we no longer need the extended
areas $P_{1-}^{\text{ext}}$ and $P_{1+}^{\text{ext}}$. The subset of the
Preisach domain $P_{1}$ is now subdivided in four disjoint regions defined by

$\displaystyle P_{1}^{1}$ $\displaystyle:=$ $\displaystyle\\{(\alpha,\beta)\
|\ \alpha\leq 0,\ \beta\leq 0\\},$ $\displaystyle P_{1}^{2}$ $\displaystyle:=$
$\displaystyle\\{(\alpha,\beta)\ |\ \alpha>0,\ \beta\leq 0,\
\alpha\leq-\beta\\},$ $\displaystyle P_{1}^{3}$ $\displaystyle:=$
$\displaystyle\\{(\alpha,\beta)\ |\ \beta\leq 0,\ \alpha>-\beta\\},$
$\displaystyle P_{1}^{4}$ $\displaystyle:=$ $\displaystyle\\{(\alpha,\beta)\
|\ \alpha>0,\ \beta>0\\}.$

(a) Input signal (b) Output signal (c) Input-output phase plot

Figure 6: Input-output phase plot using the Preisach butterfly hysteresis
operator with symmetric two-sided weighting function. (a). The plot of input
signal $u$ in a periodic time interval $[t_{1},t_{5}]$. (b). The corresponding
plot of output signal $y$. (c). The input-output phase plot of input and
output signal which shows a symmetric butterfly loop and whose signed-area is
equal to zero.

Note that the output can be determined by the individual behavior of each
region in the form

$\displaystyle\big{(}$ $\displaystyle\mathcal{P}(u,L_{0})\big{)}(t):=$ (8)
$\displaystyle-\iint\displaylimits_{(\alpha,\beta)\in P_{1}^{1}}\
\big{(}\mathcal{R}^{\circlearrowleft}_{\alpha,\beta}(u,r_{\alpha,\beta}(L_{0}))\big{)}(t)\
{\rm d}\hbox{\hskip 0.5pt}\alpha{\rm d}\hbox{\hskip 0.5pt}\beta$
$\displaystyle-\iint\displaylimits_{(\alpha,\beta)\in P_{1}^{2}}\
\big{(}\mathcal{R}^{\circlearrowleft}_{\alpha,\beta}(u,r_{\alpha,\beta}(L_{0}))\big{)}(t)\
{\rm d}\hbox{\hskip 0.5pt}\alpha{\rm d}\hbox{\hskip 0.5pt}\beta$
$\displaystyle+\iint\displaylimits_{(\alpha,\beta)\in P_{1}^{3}}\
\big{(}\mathcal{R}^{\circlearrowleft}_{\alpha,\beta}(u,r_{\alpha,\beta}(L_{0}))\big{)}(t)\
{\rm d}\hbox{\hskip 0.5pt}\alpha{\rm d}\hbox{\hskip 0.5pt}\beta$
$\displaystyle+\iint\displaylimits_{(\alpha,\beta)\in P_{1}^{4}}\
\big{(}\mathcal{R}^{\circlearrowleft}_{\alpha,\beta}(u,r_{\alpha,\beta}(L_{0}))\big{)}(t)\
{\rm d}\hbox{\hskip 0.5pt}\alpha{\rm d}\hbox{\hskip 0.5pt}\beta$

Let us analyze the input-output behavior when a periodic input $u$ with period
$T$ and with one maximum $u_{\max}=\beta_{1}$ and one minimum
$u_{\min}=-\beta_{1}$ is applied to this operator. Consider five time
instances $t_{1}<t_{2}<t_{3}<t_{4}<t_{5}$ with $t_{1}\geq T$ and such that
$u(t_{1})=u_{\min}=-\beta_{1}$, $u(t_{2})=0$, $u(t_{3})=u_{\max}=\beta_{1}$,
$u(t_{4})=0$, and $u(t_{5})=u_{\min}=-\beta_{1}$. It is clear that $u(t)$ is
monotonically increasing in the interval $t_{1}\leq t<t_{3}$ and monotonically
decreasing in the interval $t_{3}\leq t<t_{5}$. An example of an input signal
$u$ satisfying these conditions is illustrated in Fig. 6(a). Using such input
signal, the output signal $y(t)$ can be computed analytically for every
$t\in[t_{1},t_{5}]$ (i.e. for one periodic interval when the phase plot forms
a hysteresis loop $\mathcal{H}_{u,y}$) based on (8) and is given by

$\displaystyle\big{(}\mathcal{P}(u,L_{0})\big{)}(t)=$ (9)
$\displaystyle\left\\{\begin{array}[]{ll}-\big{(}u_{\max}+u(t)\big{)}^{2}&\text{if
}t_{1}\leq t<t_{2}\\\
-\big{(}u_{\max}-u(t)\big{)}\big{(}u_{\max}+3u(t)\big{)}&\text{if }t_{2}\leq
t<t_{3}\\\ -\big{(}u_{\max}-u(t)\big{)}^{2}&\text{if }t_{3}\leq t<t_{4}\\\
-\big{(}u_{\max}+u(t)\big{)}\big{(}u_{\max}-3u(t)\big{)}&\text{if }t_{4}\leq
t<t_{5}\end{array}\right.$

Fig. 6(b) shows the corresponding output signal. By plotting the phase plot as
in Fig. 6(c), we can see immediately that the resulting butterfly loop is
symmetric as expected. Furthermore using (1) it can be validated that the
signed-area enclosed by this curve is equal to zero.

As a final remark of this section, note that the assumption of the boundary
$B$ that separates the polar regions of the weighting function being
monotonically decreasing is made to simplify the analysis in Proposition
III.2. Such assumption guarantees that it is always possible to find the
extended domains $P_{1+}^{\text{ext},\lambda}$ or
$P_{1-}^{\text{ext},\lambda}$ where the weighting function is sign-definite.
However, according to Definition II.6 and Lemma III.1, we have that
$\mathcal{P}$ is a Preisach butterfly operator as long as we can find a
hysteresis loop $\mathcal{H}_{u,y}$ with an input $u$ whose minimum and
maximum can parameterize a subdomain $P_{1}$ of the form
$P_{1}:=\left\\{(\alpha,\beta)\in P\ |\
u_{\min}\leq\beta\leq\alpha,\,u_{\min}\leq\alpha\leq u_{\max}\right\\}$ which
satisfies

$\displaystyle\iint\displaylimits_{(\alpha,\beta)\in P_{1}}\mu(\alpha,\beta)\
\big{(}\alpha-\beta\big{)}{\rm d}\hbox{\hskip 0.5pt}{\alpha}{\rm
d}\hbox{\hskip 0.5pt}\beta=0.$

It follows that the monotonically decreasing property of the boundary $B$ is
not necessary to obtain a Preisach butterfly operator.

##  IV Preisach Multi-loop Operator

In the previous section, we have shown that a butterfly hysteresis operator
can be obtained from a Preisach operator with a two-sided weighting function.
Following from the condition of zero total signed-area in Definition II.6, the
previous analysis is based on finding an input $u$ such that each subloop
contribution to the total signed-area of the hysteresis loop canceled each
other. This analysis exploits the particular two-sided structure of the
weighting function $\mu$. However, imposing this structure to the weighting
function $\mu$ is only a sufficient condition to obtain a butterfly hysteresis
operator which, in addition, restricts all the hysteresis loops obtained from
the Preisach operator to have at most two subloops.

In this section we study a larger class of Preisach operators with more
complex weighting functions that are not necessarily two-sided. The Preisach
operators in this class can produce hysteresis loops with more than two
subloops. Therefore, using an analysis based only on the the total enclosed
signed-area of the hysteresis loops is no longer applicable for studying this
class of hysteresis operators since the signed-area does not determine
directly the number of subloops in a given hysteresis loop. In this case, we
must note that if a hysteresis loop has two or more subloops each one of the
subloops is connected to another subloop by at least one self-intersection
point of the hysteresis loop. We will call these points the crossover points
of a hysteresis loop and characterize them as follows. Consider a hysteresis
loop $\mathcal{H}_{u,y}$ obtained from an input-output pair $(u,y)$ of a
hysteresis operator $\Phi$ with $y=\Phi(u)$. Let us select $t_{1}\geq t_{p}$
such that $t_{1}<t_{2}<t_{1}+T$ is a monotone partition of one periodic
interval of $u$ and $u(t)$ is monotonically increasing when
$t\in[t_{1},t_{2}]$ and monotonically decreasing when $t\in[t_{2},t_{1}+T]$.
We can split the hysteresis loop $\mathcal{H}_{u,y}$ into two segments that
correspond to the subintervals of the monotone partition given by

$\displaystyle\mathcal{H}_{u,y}^{+}$ $\displaystyle:=\\{(u(t),y(t))\ |\
t\in[t_{1},t_{2}]\\},$ (10) $\displaystyle\mathcal{H}_{u,y}^{-}$
$\displaystyle:=\\{(u(t),y(t))\ |\ t\in[t_{2},t_{1}+T]\\}.$ (11)

We define formally a crossover point as follows.

###### Definition IV.1

Consider a hysteresis loop $\mathcal{H}_{u,y}$. A point
$(u_{c},y_{c})\in\mathcal{H}_{u,y}$ is called a crossover point if
$(u_{c},y_{c})\in\mathcal{H}_{u,y}^{+}\cap\mathcal{H}_{u,y}^{-}$.
$\hfill\triangle$

We remark from the definition above that a hysteresis loop $\mathcal{H}_{u,y}$
will always have at least two crossover points corresponding to the points
where the input $u$ achieves its extrema
$(u_{\min},y_{1}),(u_{\max},y_{2})\in\mathcal{H}_{u,y}$ with
$(u_{\min},y_{1})=(u(t_{1}),y(t_{1}))=(u(t_{1}+T),y(t_{1}+T))$ and
$(u_{\max},y_{2})=(u(t_{2}),y(t_{2}))$. Moreover, it is possible for a
hysteresis loop $\mathcal{H}_{u,y}$ to have an infinite number of crossover
points if, for instance, there exists a segment of intersection between
$\mathcal{H}_{u,y}^{+}$ and $\mathcal{H}_{u,y}^{-}$, i.e. there exists a time
subinterval $[t_{3},t_{4}]\subset[t_{1},t_{1}+T]$ with $t_{4}>t_{3}$ such that
$(y(t),u(t))\in\mathcal{H}_{u,y}^{+}\cap\mathcal{H}_{u,y}^{-}$ for every
$t\in[t_{3},t_{4}]$. For this reason, it can be checked that the numbers of
subloops in a hysteresis loop $\mathcal{H}_{u,y}$ is not determined by the
number of crossover points but by the number of maximal connected subsets in
$\mathcal{H}_{u,y}^{+}\cap\mathcal{H}_{u,y}^{-}$, where each maximal connected
subset can be a singleton in the case that the corresponding crossover point
does not belong to a segment of intersection between $\mathcal{H}_{u,y}^{+}$
and $\mathcal{H}_{u,y}^{-}$. By a maximal connected subset in $A$ we mean a
connected subset $B\subset A$ with the property that there does not exist
other connected subset $C\subset A$ such that $B\subset C$.

Using these notions we introduce the definition of multi-loop hysteresis
operator as follows.

###### Definition IV.2

A hysteresis operator $\Phi$ is called a multi-loop hysteresis operator if
there exists a hysteresis loop $\mathcal{H}_{u,y}$, where $y=\Phi(u)$, with at
least one maximal connected subset
$C\subset\mathcal{H}_{u,y}^{+}\cap\mathcal{H}_{u,y}^{-}$ such that
$u_{\min}\neq u_{c}\neq u_{\max}$ for every $(u_{c},y_{c})\in C$.
$\hfill\triangle$

Definition IV.2 is asking for the existence of at least one maximal connected
subset of crossover points besides the ones that contain the crossover points
corresponding to the maximum and minimum values of the input. The existence of
this maximal connected subset guarantees that the hysteresis loop will
composed of at least two subloops. Moreover, it is clear that every butterfly
hysteresis operator is then a multi-loop hysteresis operator. Before
characterizing the class of Preisach operators that satisfy the conditions to
be multi-loop hysteresis operators, we introduce next lemma that allows us to
relate the existence of a crossover point in a hysteresis loop obtained from a
Preisach operator with the integration of its weighting function $\mu$ over a
rectangular region of $P$ delimited by the maximum and minimum values of the
input.

###### Lemma IV.3

Consider a hysteresis loop $\mathcal{H}_{u,y}$ obtained from an input-output
pair $(u,y)$ of a Preisach operator $\mathcal{P}$ with a weighting function
$\mu$. A point $(u_{c},y_{c})\in\mathcal{H}_{u,y}$ is a crossover point if and
only if

$\iint\displaylimits_{(\alpha,\beta)\in\Omega_{c}}\mu(\alpha,\beta)\ {\rm
d}\hbox{\hskip 0.5pt}\alpha{\rm d}\hbox{\hskip 0.5pt}\beta=0$ (12)

where the region $\Omega_{c}$ is defined by $\Omega_{c}:=\\{(\alpha,\beta)\in
P\ |\ u_{c}<\alpha<u_{\max},\ u_{\min}<\beta<u_{c}\\}$.

###### Proof:

Lemma IV.3 (Sufficiency) Let $(u_{c},y_{c})$ be the crossover point of a
hysteresis loop $\mathcal{H}_{u,y}$ and consider the corresponding subsets
$\mathcal{H}_{u,y}^{+}$ and $\mathcal{H}_{u,y}^{-}$ of $\mathcal{H}_{u,y}$ as
defined in (10) and (11) with a monotone partition $t_{1}<t_{2}<t_{1}+T$ of
the periodic input $u$. When $u_{c}=u_{\min}=u(t_{1})=u(t_{1}+T)$ or
$u_{c}=u_{\max}=u(t_{2})$, the region $\Omega_{c}$ is empty and (12) holds
trivially. Therefore, let us consider the case when there exist two time
instants $\tau_{1}\in(t_{1},t_{2})$ and $\tau_{2}\in(t_{2},t_{1}+T)$ such that
$(u_{c},y_{c})=(u(\tau_{1}),y(\tau_{1}))=(u(\tau_{2}),y(\tau_{2}))$ with
$u_{\min}\neq u_{c}\neq u_{\max}$.

Let us analyze the input-output behavior of the Preisach operator in the
intervals $(t_{1},t_{2})$ and $(t_{1},t_{2}+T)$. For this, consider a
subdomain of the Preisach plane given by $P_{1}:=\\{(\alpha,\beta)\in P\ |\
\alpha<u_{\max},\ \beta>u_{\min}\\}$ which is a triangle whose vertices are at
$(u_{\max},u_{\min})$, $(u_{\max},u_{\max})$ and $(u_{\min},u_{\min})$. It is
clear that at every time instance $t\geq t_{p}=T$ the state of relays in
$P\backslash P_{1}$ remains the same as given by the initial condition. We
define three time varying disjoint regions of $P_{1}$ whose boundaries depend
on the instantaneous value of the input $u(t)$ and which are given by

$\displaystyle\Omega_{1}(t)$ $\displaystyle:=\left\\{(\alpha,\beta)\in P_{1}\
|\ u(t)<\beta\right\\},$ (13) $\displaystyle\Omega_{2}(t)$
$\displaystyle:=\left\\{(\alpha,\beta)\in P_{1}\ |\
\beta<u(t)<\alpha\right\\},$ (14) $\displaystyle\Omega_{3}(t)$
$\displaystyle:=\left\\{(\alpha,\beta)\in P_{1}\ |\ \alpha<u(t)\right\\}.$
(15)

The region $\Omega_{1}(t)$ is a triangle whose vertices are at $(u(t),u(t))$,
$(u_{\max},u(t))$ and $(u_{\max},u_{\max})$, the region $\Omega_{3}(t)$ is a
triangle whose vertices are at $(u(t),u(t))$, $(u(t),u_{\min})$ and
$(u_{\min},u_{\min})$, and the region $\Omega_{2}(t)$ is a rectangle whose
vertices are at $(u(t),u(t))$, $(u_{\max},u_{\min})$, $(u(t),u_{\min})$ and
$(u_{\max},u(t))$. It can be checked that for every time instance
$t\in[t_{1},t_{1}+T]$, all relays corresponding to the regions $\Omega_{1}(t)$
and $\Omega_{3}(t)$ are in state $-1$ and $+1$, respectively. Moreover, at
time instances $\tau_{1}$ and $\tau_{2}$ we have that
$\Omega_{2}(\tau_{1})=\Omega_{2}(\tau_{2})=\Omega_{c}$.

The required condition (12) is obtained computing the output of the Preisach
operator at both time instances $\tau_{1}$ and $\tau_{2}$ using the regions
$\Omega_{1}(t)$, $\Omega_{2}(t)$ and $\Omega_{3}(t)$ as follows. At time
instance $t_{1}$ the input $u$ reaches its minimum value. Thus we have
$u(t_{1})=u_{\min}$ which implies that all relays in the subdomain $P_{1}$ are
in $-1$ state. As the input increases, at every time instance
$t\in(t_{1},t_{2})$ the region $\Omega_{3}(t)$ indicates the relays whose
state has changed from $-1$ to $+1$ while in the regions $\Omega_{2}(t)$ and
$\Omega_{1}(t)$ all relays remain in $-1$ state. Therefore, at time instance
$\tau_{1}$ the output of the Preisach operator is given by

$\displaystyle y(\tau_{1})=\big{(}\mathcal{P}(u,L_{0})\big{)}(\tau_{1})$ (16)
$\displaystyle-\iint\displaylimits_{(\alpha,\beta)\in\Omega_{1}(\tau_{1})}\
\mu(\alpha,\beta)\ {\rm d}\hbox{\hskip 0.5pt}\alpha{\rm d}\hbox{\hskip
0.5pt}\beta\ \ -\iint\displaylimits_{(\alpha,\beta)\in\Omega_{2}(\tau_{1})}\
\mu(\alpha,\beta)\ {\rm d}\hbox{\hskip 0.5pt}\alpha{\rm d}\hbox{\hskip
0.5pt}\beta$
$\displaystyle+\iint\displaylimits_{(\alpha,\beta)\in\Omega_{3}(\tau_{1})}\
\mu(\alpha,\beta)\ {\rm d}\hbox{\hskip 0.5pt}\alpha{\rm d}\hbox{\hskip
0.5pt}\beta$ $\displaystyle+\iint\displaylimits_{(\alpha,\beta)\in P\backslash
P_{1}}\
\mu(\alpha,\beta)\big{(}\mathcal{R}^{\circlearrowleft}_{\alpha,\beta}(u,r_{\alpha,\beta}(L_{0}))\big{)}(\tau_{1})\
{\rm d}\hbox{\hskip 0.5pt}\alpha{\rm d}\hbox{\hskip 0.5pt}\beta.$

At time instance $t_{2}$ the input $u$ reaches now its maximum value. Thus, in
this case we have $u(t_{2})=u_{\max}$ which implies that all relays in the
subdomain $P_{1}$ are in $+1$ state. Subsequently, as the input decreases, at
every time instance $t\in(t_{2},t_{1}+T)$ the region $\Omega_{1}(t)$ indicates
the relays whose state has changed from $+1$ to $-1$ while in the regions
$\Omega_{2}(t)$ and $\Omega_{3}(t)$ all relays remain in $+1$ state.
Therefore, at time instance $\tau_{2}$ the output of the Preisach operator is
given by

$\displaystyle y(\tau_{2})=\big{(}\mathcal{P}(u,L_{0})\big{)}(\tau_{2})$ (17)
$\displaystyle-\iint\displaylimits_{(\alpha,\beta)\in\Omega_{1}(\tau_{2})}\
\mu(\alpha,\beta)\ {\rm d}\hbox{\hskip 0.5pt}\alpha{\rm d}\hbox{\hskip
0.5pt}\beta\ \ +\iint\displaylimits_{(\alpha,\beta)\in\Omega_{2}(\tau_{2})}\
\mu(\alpha,\beta)\ {\rm d}\hbox{\hskip 0.5pt}\alpha{\rm d}\hbox{\hskip
0.5pt}\beta$
$\displaystyle+\iint\displaylimits_{(\alpha,\beta)\in\Omega_{3}(\tau_{2})}\
\mu(\alpha,\beta)\ {\rm d}\hbox{\hskip 0.5pt}\alpha{\rm d}\hbox{\hskip
0.5pt}\beta$ $\displaystyle+\iint\displaylimits_{(\alpha,\beta)\in P\backslash
P_{1}}\
\mu(\alpha,\beta)\big{(}\mathcal{R}^{\circlearrowleft}_{\alpha,\beta}(u,r_{\alpha,\beta}(L_{0}))\big{)}(\tau_{2})\
{\rm d}\hbox{\hskip 0.5pt}\alpha{\rm d}\hbox{\hskip 0.5pt}\beta.$

Subtracting (17) and (16) we have

$\displaystyle 0$ $\displaystyle=y(\tau_{2})-y(\tau_{1})$ (18)
$\displaystyle=\iint\displaylimits_{(\alpha,\beta)\in\Omega_{2}(\tau_{2})}\
\mu(\alpha,\beta)\ {\rm d}\hbox{\hskip 0.5pt}\alpha{\rm d}\hbox{\hskip
0.5pt}\beta+\iint\displaylimits_{(\alpha,\beta)\in\Omega_{2}(\tau_{1})}\
\mu(\alpha,\beta)\ {\rm d}\hbox{\hskip 0.5pt}\alpha{\rm d}\hbox{\hskip
0.5pt}\beta$
$\displaystyle=2\iint\displaylimits_{(\alpha,\beta)\in\Omega_{c}}\
\mu(\alpha,\beta)\ {\rm d}\hbox{\hskip 0.5pt}\alpha{\rm d}\hbox{\hskip
0.5pt}\beta.$

(Necessity) Assume that (12) holds for some value
$u_{c}\in[u_{\min},u_{\max}]$. Consider the time instance
$\tau_{1}\in[t_{1},t_{2}]$ when $u(\tau_{1})=u_{c}$. At this time instance,
the output $y(\tau_{1})$ is given as in (16) and
$(u(\tau_{1}),y(\tau_{1}))\in\mathcal{H}_{u,y}^{+}$. Similarly, let
$\tau_{2}\in[t_{2},t_{1}+T]$ be the time instance when $u(\tau_{2})=u_{c}$. At
this time instance the output $y(\tau_{2})$ is given as in (17) and
$(u(\tau_{2}),y(\tau_{2}))\in\mathcal{H}_{u,y}^{-}$. Since (12) holds, by
subtracting (17) and (16) we obtain (18) again. It follows that
$y(\tau_{2})=y(\tau_{1})=y_{c}$, and consequently
$(u_{c},y_{c})\in\mathcal{H}_{u,y}^{+}\cap\mathcal{H}_{u,y}^{-}$. ∎

It is clear that Lemma IV.3 can be used to find the crossover points of a
hysteresis loop obtained from a Preisach operator. However, noting that the
region $\Omega_{c}$ in (12) depends explicitly on the maximum and minimum of
the input $u$ applied to the Preisach operator, we could also use Lemma IV.3
to estimate an input $u$ that produces a hysteresis loop with crossover points
additional to the trivial ones corresponding to the maximum and minimum value
of the input.

###### Example IV.4

Let us recall the Preisach operator from Example III.4 whose weighting
function $\mu$ is defined in (7). We can check that a non-empty region
$\Omega_{c}$ satisfying (12) is given by

$\Omega_{c}=\\{(\alpha,\beta)\in P\ |\ 0<\alpha<\beta_{1},\
-\beta_{1}<\beta<0\\},$ (19)

which is illustrated in Fig. 7. Therefore, noting the limits of region
$\Omega_{c}$ in (19), it follows that applying to this Preisach operator an
input $u$ with one maximum $u_{\max}=\beta_{1}$ and one minimum
$u_{\min}=-\beta_{1}$ yields a crossover point $(u_{c},y_{c})$ with $u_{c}=0$
as it has been shown in the phase plot of Fig 6. Furthermore, using (9) we can
find that $y_{c}=-u_{\max}^{2}$.

Figure 7: Weighting function $\mu(\alpha,\beta)$ of Example III.4 defined in
(7) where the region $\Omega_{c}$ given by (7) and that satisfies (12) is
indicated by the dashed line.

We remark that only the existence of crossover points does not guarantee that
the hysteresis loop is composed of subloops with different orientation. We
illustrate this with the next example.

###### Example IV.5

Consider a subdomain of Preisach plane $P_{1}=\\{(\alpha,\beta)\in P\ |\
\alpha<\beta_{1},\,\beta>-\beta_{1}\\}$ with $\beta_{1}>0$ and define a
weighting function $\mu$ given by

$\displaystyle\mu(\alpha,\beta)$
$\displaystyle:=\left\\{\begin{array}[]{rl}-1,&\text{if }(\alpha,\beta)\in
P_{1-},\\\ 1,&\text{if }(\alpha,\beta)\in P_{1}\backslash P_{1-},\\\
0,&\text{otherwise}.\end{array}\right.$ (20)

where $P_{1-}=\\{(\alpha,\beta)\in P_{1}\ |\
-\beta_{1}<\beta<0,\,0<\alpha<\beta+\beta_{1}\\}$. It can be checked that the
same region $\Omega_{c}$ given as in (19) satisfies condition (12) with the
weighting function $\mu$ defined by (20). Fig. 8 illustrates this weighting
function with the region $\Omega_{c}$ indicated by a dashed line. It follows
that the hysteresis loop obtained from a Preisach operator with a weighting
function $\mu$ defined by (20) and whose input $u$ has one maximum
$u_{\max}=\beta_{1}$ and one minimum $u_{\min}=-\beta_{1}$ has a crossover
point with coordinates $(u_{c},y_{c})=(0,0)$, where $y_{c}$ is computed using
(16) or (17). Nevertheless, by simple inspection of the phase plot include in
Fig. 9, we can check that the hysteresis loop is composed of two subloops with
the same orientation.

Figure 8: Weighting function $\mu(\alpha,\beta)$ of a Preisach operator which
can produce a hysteresis loop whose subloops have the same orientation. The
region $\Omega_{c}$ that satisfies (12) is indicated by the dashed line.

(a) Input signal (b) Output signal (c) Input-output phase plot

Figure 9: Input-output hysteresis response using the multiple loops Preisach
multi-loop operator. (a). The plot of input signal $u$ in a periodic time
interval $[t_{1},t_{1}+T]$. (b). The corresponding plot of output signal $y$.
(c). The input-output phase plot of input and output signal which shows
hysteresis loop with two subloops in the same orientation.

We introduce now a proposition that shows how a multi-loop hysteresis operator
can be obtained from a Preisach operator.

###### Proposition IV.6

Consider a Preisach operator $\mathcal{P}$ as in
$\eqref{eq:preisach_operator}$ with a weighting function $\mu$. Assume that
there exists a point $(\alpha_{0},\beta_{0})\in P$ with $\alpha_{0}>\beta_{0}$
and three values $\alpha_{1^{-}}<\alpha_{1}<\alpha_{1^{+}}$ such that
$\beta_{0}<\alpha_{1^{-}}$ and $\alpha_{1^{+}}<\alpha_{0}$, and (12) holds for
the region

$\Omega_{\alpha_{1}}=\\{(\alpha,\beta)\in P\ |\ \alpha_{1}<\alpha<\alpha_{0},\
\beta_{0}<\beta<\alpha_{1}\\}$ (21)

but does not hold for the regions

$\displaystyle\Omega_{\alpha_{1^{-}}}$ $\displaystyle=\\{(\alpha,\beta)\in P\
|\ \alpha_{1^{-}}<\alpha<\alpha_{0},\ \beta_{0}<\beta<\alpha_{1^{-}}\\},$ (22)
$\displaystyle\Omega_{\alpha_{1^{+}}}$ $\displaystyle=\\{(\alpha,\beta)\in P\
|\ \alpha_{1^{+}}<\alpha<\alpha_{0},\ \beta_{0}<\beta<\alpha_{1^{+}}\\}.$ (23)

Then $\mathcal{P}$ is a multi-loop hysteresis operator.

###### Proof:

Proposition IV.6 Consider the hysteresis loop $\mathcal{H}_{u,y}$ obtained
from the input-output pair $(u,y)$ with the input $u$ being periodic with one
maximum $u_{\max}=\alpha_{0}$ and one minimum $u_{\min}=\beta_{0}$, and
$y=\mathcal{P}(u,L_{0})$. Let $t_{1}<t_{2}<t_{1}+T$ be the monotonic partition
of the input that divides $\mathcal{H}_{u,y}$ into $\mathcal{H}_{u,y}^{-}$ and
$\mathcal{H}_{u,y}^{+}$ with $u(t_{1})=u(t_{1}+T)=u_{\min}$ and
$u(t_{2})=u_{\max}$, and consider six time instances
$\tau_{1^{-}},\tau_{1},\tau_{1^{+}}\in[t_{1},t_{2}]$ and
$\tau_{2^{-}},\tau_{2},\tau_{2^{+}}\in[t_{2},t_{1}+T]$ with
$\tau_{1^{-}}<\tau_{1}<\tau_{1^{+}}$ and $\tau_{2^{-}}>\tau_{2}>\tau_{2^{+}}$
such that $u(\tau_{1^{-}})=u(\tau_{2^{-}})=\alpha_{1^{-}}$,
$u(\tau_{1})=u(\tau_{2})=\alpha_{1}$ and
$u(\tau_{1^{+}})=u(\tau_{2^{+}})=\alpha_{1^{+}}$.

Using Lemma IV.3 with the region $\Omega_{\alpha_{1}}$ defined in (21), the
hysteresis loop $\mathcal{H}_{u,y}$ has a crossover point $(u_{c},y_{c})$ with
$u_{c}=u(\tau_{1})=u(\tau_{2})=\alpha_{1}$ and where
$y_{c}=y(\tau_{1})=y(\tau_{2})$ is given by (16) or (17).

Without loss of generality, let $C$ be the maximal connected subset of
$\mathcal{H}_{u,y}^{+}\cap\mathcal{H}_{u,y}^{-}$ that contains
$(u_{c},y_{c})$. To check that $C$ does not contain a crossover point of the
form $(u_{\min},y_{1})$ observe that since (12) does not hold for the region
$\Omega_{\alpha_{1^{-}}}$ then using again Lemma IV.3 we have that
$(u(\tau_{1^{-}}),y(\tau_{1^{-}}))$ and $(u(\tau_{2^{-}}),y(\tau_{2^{-}}))$
are not crossover points and are not in
$\mathcal{H}_{u,y}^{+}\cap\mathcal{H}_{u,y}^{-}$. Consequently, there does not
exist connected subset of $\mathcal{H}_{u,y}^{+}\cap\mathcal{H}_{u,y}^{-}$
that could contain both $(u_{c},y_{c})$ and $(u_{\min},y_{1})$. Similarly, we
can check that $C$ does not contain a crossover point of the form
$(u_{\max},y_{2})$ by noting that (12) does not hold for the region
$\Omega_{\alpha_{1^{+}}}$ which by Lemma IV.3 implies that
$(u(\tau_{1^{+}}),y(\tau_{1^{+}}))$ and $(u(\tau_{2^{+}}),y(\tau_{2^{+}}))$
are not crossover points and are not in
$\mathcal{H}_{u,y}^{+}\cap\mathcal{H}_{u,y}^{-}$. It follows again that there
does not exist connected subset of
$\mathcal{H}_{u,y}^{+}\cap\mathcal{H}_{u,y}^{-}$ that could contain both
$(u_{c},y_{c})$ and $(u_{\max},y_{2})$. ∎

We remark that Proposition IV.6 could be extended to consider more than one
region $\Omega_{\alpha}$ where (12) holds. Let $\mu$ be a weighting function
and consider values

$\alpha_{i^{-}}<\alpha_{i}<\alpha_{i^{+}},\quad\text{where
}i\in\left\\{1,\dots,m\right\\}\text{ and }m\in\mathbb{Z}_{+},$

with $\beta_{0}<\alpha_{1^{-}}$ and $\alpha_{m^{+}}<\alpha_{0}$, and such that
for every $j\in\left\\{1,\dots,m-1\right\\}$ we have that
$\alpha_{j^{+}}<\alpha_{(j+1)^{-}}$. Assume that using these values, we can
construct regions given by

$\Omega_{\alpha_{i}}=\\{(\alpha,\beta)\in P\ |\ \alpha_{i}<\alpha<\alpha_{0},\
\beta_{0}<\beta<\alpha_{i}\\},$

such that (12) holds but does not hold for regions given by

$\displaystyle\Omega_{\alpha_{i^{-}}}$ $\displaystyle=\\{(\alpha,\beta)\in P\
|\ \alpha_{i^{-}}<\alpha<\alpha_{0},\ \beta_{0}<\beta<\alpha_{i^{-}}\\},$
$\displaystyle\Omega_{\alpha_{i^{+}}}$ $\displaystyle=\\{(\alpha,\beta)\in P\
|\ \alpha_{i^{+}}<\alpha<\alpha_{0},\ \beta_{0}<\beta<\alpha_{i^{+}}\\}$

for every $i\in\left\\{1,\dots,m\right\\}$. It follows immediately from
Proposition IV.6 that a Preisach operator with this weighting function is a
multi-loop hysteresis operator. Moreover, it can be checked that in this case
the hysteresis loop $\mathcal{H}_{u,y}$ obtained from such Preisach operator
with an input whose maximum is $u_{\max}=\alpha_{0}$ and minimum is
$u_{\min}=\beta_{0}$ will be composed of $m+1$ subloops. The final example of
this section illustrates a Preisach operator with a weighting function that
has a complex distribution of positive and negative domains and whose
hysteresis loops has four subloops.

###### Example IV.7

Consider a subset of Preisach domain $P_{1}:=\left\\{(\alpha,\beta)\ |\
-1<\beta<1,\ \beta<\alpha<1\right\\}$ and a weighting function defined by

$\displaystyle\mu(\alpha,\beta):=$ (24)
$\displaystyle\left\\{\begin{array}[]{c@{\hspace{0.25cm}}l}\sin\left(2\pi\left(\alpha-\beta\right)\right)+\sin\left(2\pi\left(\alpha+\beta\right)\right),\hfil\hskip
7.11317pt&\text{if }(\alpha,\beta)\in P_{1},\\\ 0,\hfil\hskip
7.11317pt&\text{otherwise}.\end{array}\right.$

Figure 10: Weighting function $\mu(\alpha,\beta)$ defined in (24)
corresponding to a Preisach multi-loop operator.

The weighting function $\mu$ defined in (24) is illustrated in Fig 10. For
this weighting function there exist three non-empty regions
$\Omega_{\alpha_{1}}$, $\Omega_{\alpha_{2}}$ and $\Omega_{\alpha_{3}}$ that
satisfy (12) and which are given by

$\displaystyle\Omega_{\alpha_{1}}$ $\displaystyle=\\{(\alpha,\beta)\in P_{1}\
|\ -0.5<\alpha<1,\ -1<\beta<-0.5\\},$ (25) $\displaystyle\Omega_{\alpha_{2}}$
$\displaystyle=\\{(\alpha,\beta)\in P_{1}\ |\ 0<\alpha<1,\ -1<\beta<0\\},$
$\displaystyle\Omega_{\alpha_{3}}$ $\displaystyle=\\{(\alpha,\beta)\in P_{1}\
|\ 0.5<\alpha<1,\ -1<\beta<0.5\\}.$

Moreover, it can be verified that (12) does not hold for every of next the
regions

$\displaystyle\Omega_{\alpha_{1^{-}}}$ $\displaystyle=\\{(\alpha,\beta)\in
P_{1}\,|\,-0.75<\alpha<1,-1<\beta<-0.75\\},$
$\displaystyle\Omega_{\alpha_{1^{+}}}$ $\displaystyle=\\{(\alpha,\beta)\in
P_{1}\,|\,-0.25<\alpha<1,-1<\beta<-0.25\\},$
$\displaystyle\Omega_{\alpha_{2^{-}}}$ $\displaystyle=\\{(\alpha,\beta)\in
P_{1}\,|\,-0.25<\alpha<1,-1<\beta<-0.25\\},$
$\displaystyle\Omega_{\alpha_{2^{+}}}$ $\displaystyle=\\{(\alpha,\beta)\in
P_{1}\,|\,0.25<\alpha<1,-1<\beta<0.25\\},$
$\displaystyle\Omega_{\alpha_{3^{-}}}$ $\displaystyle=\\{(\alpha,\beta)\in
P_{1}\,|\,0.25<\alpha<1,-1<\beta<0.25\\},$
$\displaystyle\Omega_{\alpha_{3^{+}}}$ $\displaystyle=\\{(\alpha,\beta)\in
P_{1}\,|\,0.75<\alpha<1,-1<\beta<0.75\\}.$

Fig. 12 illustrates the three regions $\Omega_{\alpha_{1}}$,
$\Omega_{\alpha_{2}}$ and $\Omega_{\alpha_{3}}$ with a dashed line and Fig. 11
shows the input-output phase plot of the Preisach operator with the weighting
function (24) with a periodic input whose maximum and minimum are
$u_{\min}=-1$ and $u_{\max}=1$. It can be verified that the hysteresis loop is
composed of four subloops and that there exist three crossover points
additional to the trivial ones corresponding to the maximum and minimum of the
input.

(a) Input signal (b) Output signal (c) Input-output phase plot

Figure 11: Input-output hysteresis response using the multiple loops Preisach
multi-loop operator. (a). The plot of input signal $u$ in a periodic time
interval $[t_{1},t_{1}+T]$. (b). The corresponding plot of output signal $y$.
(c). The input-output phase plot of input and output signal which shows the
hysteresis loop with more than two subloops in different orientations.

Figure 12: Weighting function $\mu(\alpha,\beta)$ defined in (24)
corresponding to a Preisach multi-loop operator. The regions defined in (25)
that satisfy (12) are indicated by the dashed line: $\Omega_{\alpha_{1}}$ in
(a), $\Omega_{\alpha_{2}}$ in (b), and $\Omega_{\alpha_{3}}$ in (c).

##  V Set stability of a Lur’e system with a Preisach multi-loop operator in
the feedback loop

In this section, we present a brief study of the stability of a Lur’e-type
system where the nonlinearity in the feedback loop is described by a Preisach
multi-loop operator. We based our analysis on the results introduced in [20]
where a bounded relation between the input and output rate of the Preisach
operator has been found. Let us consider a Lur’e system which is described by

$\displaystyle\Sigma_{1}:$ $\displaystyle\begin{aligned}
\dot{x}(t)&=Ax(t)+Bw(t),\quad x(0)=x_{0},\\\ z(t)&=Cx(t)\\\ \end{aligned}$
(26) $\displaystyle\Sigma_{2}:$ $\displaystyle
y(t)=\big{(}\mathcal{P}(u,L_{0})\big{)}(t),\quad L_{0}\in\mathcal{I},$
$\displaystyle\qquad\qquad\qquad\qquad\text{ with }w(t)=-y(t),\ u(t)=z(t),$

where $\Sigma_{1}$ is a linear system, $x(t)\in\mathbb{R}^{n}$,
$z(t),v(t),y(t)\in\mathbb{R}$ and $A,B,C$ are the system’s matrices with
suitable dimension, and transfer function of $\Sigma_{1}$ is given by
$G(s)=C(sI-A)^{-1}B$. Additionally, $\Sigma_{2}$ is a Preisach multi-loop
operator. This Lur’e system has a set of equilibria given by

$\mathcal{E}=\\{(\bar{x},\,\bar{L})\in\mathbb{R}^{n}\times\mathcal{I}\ |\
A\bar{x}-B\mathcal{P}(C\bar{x},\bar{L})=0\\}.$

Following from [20, Proposition 3.2 & 3.3], when the weighting function $\mu$
of the Preisach operator is compactly supported, the relation between the
input and output rate can be expressed by

$\dot{y}(t)=\psi(t)\dot{u}(t),\qquad\text{a.a. }t\in\mathbb{R}_{+},$

where $\lambda_{m}\leq\psi(t)\leq\lambda_{M}$ with $\lambda_{m}$ and
$\lambda_{M}$ given by

$\displaystyle\lambda_{m}$ $\displaystyle=2\
\min\left\\{\inf_{(\gamma,\kappa)\in
P}\int\displaylimits_{\kappa}^{\gamma}\mu(\gamma,\beta){\rm d}\hbox{\hskip
0.5pt}\beta,\ \inf_{(\gamma,\kappa)\in
P}\int\displaylimits_{\kappa}^{\gamma}\mu(\alpha,\kappa){\rm d}\hbox{\hskip
0.5pt}\alpha\right\\},$ $\displaystyle\lambda_{M}$ $\displaystyle=2\
\max\left\\{\sup_{(\gamma,\kappa)\in
P}\int\displaylimits_{\kappa}^{\gamma}\mu(\gamma,\beta){\rm d}\hbox{\hskip
0.5pt}\beta,\ \sup_{(\gamma,\kappa)\in
P}\int\displaylimits_{\kappa}^{\gamma}\mu(\alpha,\kappa){\rm d}\hbox{\hskip
0.5pt}\alpha\right\\}.$

We refer the interested readers to [20] for the details and proofs of these
claims. We state the next corollary which follows directly from [20,
Proposition 4.1].

###### Corollary V.1

Let $\mathcal{P}$ be the Preisach multi-loop operator with a compactly
supported $\mu$. Assume that $(A,C)$ is observable and $(A,B)$ is
controllable. Assume that $\overline{G}\left(j\omega\right)$ given by

$\overline{G}\left(j\omega\right):=\left(1+\lambda_{M}G(j\omega)\right)\left(1+\lambda_{m}G(j\omega)\right)^{-1},$

is strictly positive real with $\lambda_{M}>0$ and $\lambda_{m}<0$ being the
upper and lower bound of $\psi(t)$. Then $(x(t),L_{t})\to\mathcal{E}$ as
$t\to\infty$.

###### Example V.2

Consider a Lur’e system as defined in (26) whose linear system matrices are
given by

$\displaystyle A$ $\displaystyle=\left[\begin{matrix}0&1&0\\\ 0&0&1\\\
-26&-28&-3\end{matrix}\right],$ $\displaystyle B$
$\displaystyle=\left[\begin{matrix}0\\\ 0\\\ -26\end{matrix}\right],$
$\displaystyle C$
$\displaystyle=\left[\begin{matrix}1&0&0\end{matrix}\right].$

Let the Preisach multi-loop operator $\mathcal{P}$ in this Lur’e system have
the weighting function defined by (24) in Example IV.5. It can be checked that
$\lambda_{M}=\frac{4}{\pi}$ and $\lambda_{m}=-\frac{1}{2\pi}$. Moreover, it
can be checked that conditions of Corollary V.1 are satisfied. The results of
a simulation of this Lur’e system with initial conditions of the linear system
given by $x_{0}=\left[0.8,-1.0,-1.0\right]^{\top}$ and initial interface for
the Preisach multi-loop operator given by $L_{0}=\\{(\alpha,\beta)\in P\ |\
0<\alpha<1,\,\beta=-0.9\\}\cup\\{(\alpha,\beta)\in P\ |\
\alpha=1,\,\beta<-0.9\\}\cup\\{(\alpha,\beta)\in P\ |\
\alpha=0,\,-0.9<\beta\\}$ is illustrated in Fig. 13.

Figure 13: Results of a simulation of a Lur’e system whose nonlinearity is the
Preisach multi-loop operator with a weighting function defined as in (24). (a)
Input-output phase plot. The black dashed line shows the major hysteresis
loop, the red dashed line indicates the input-output pairs which correspond to
states $(\bar{x},\bar{L})\in\mathcal{E}$, and the simulation is indicated by
the blue line where the initial input-output $(y(0),u(0))$ is indicated by the
circle and the final input-output $(y(t_{f}),u(t_{f}))$ is marked by the
cross. (b) Linear system states.

##  VI Conclusion

In this paper we have introduced the concepts of butterfly hysteresis operator
based on the characterization of the enclosed signed-area of its hysteresis
loops and multi-loop hysteresis operator based on the self-intersections of
its hysteresis loops. We have studied the Preisach operator and provided
conditions over its weighting function such that a butterfly or a multi-loop
hysteresis operator can be obtained. Moreover, we analyzed the classical
problem of a Lur’e system using a Preisach multi-loop as feedback loop.

## References

  * [1] D. Bernstein, “Ivory ghost [ask the experts],” _Control Systems, IEEE_ , vol. 27, pp. 16 – 17, 11 2007.
  * [2] F. Preisach, “Über die magnetische nachwirkung,” _Zeitschrift für Physik A Hadrons and Nuclei_ , vol. 94, no. 5, pp. 277–302, 1935.
  * [3] I. D. Mayergoyz and G. Friedman, “Generalized Preisach model of hysteresis,” _IEEE Transactions on Magnetics_ , vol. 24, no. 1, pp. 212–217, Jan. 1988.
  * [4] A. Visintin, _Differential Models of Hysteresis_ , ser. Applied Mathematical Sciences. Springer Berlin Heidelberg, 1994.
  * [5] J. W. Macki, P. Nistri, and P. Zecca, “Mathematical models for hysteresis,” _SIAM review_ , vol. 35, no. 1, pp. 94–123, 1993.
  * [6] G.-Y. Gu, L.-M. Zhu, C.-Y. Su, H. Ding, and S. Fatikow, “Modeling and control of piezo-actuated nanopositioning stages: A survey,” _IEEE Transactions on Automation Science and Engineering_ , vol. 13, no. 1, pp. 313–332, 2016.
  * [7] M. Brokate and J. Sprekels, _Hysteresis and Phase Transitions_. New York: Springer-Verlag, 1996.
  * [8] R. Iyer, X. Tan, and P. Krishnaprasad, “Approximate inversion of the Preisach hysteresis operator with application to control of smart actuators,” _IEEE Trans. Automatic Control_ , vol. 50, no. 6, pp. 798–810, 2005.
  * [9] B. Jayawardhana, R. Ouyang, and V. Andrieu, “Stability of systems with the Duhem hysteresis operator: The dissipativity approach,” _Automatica_ , vol. 48, no. 10, pp. 2657–2662, Oct. 2012.
  * [10] R. Ouyang, V. Andrieu, and B. Jayawardhana, “On the characterization of the Duhem hysteresis operator with clockwise input–output dynamics,” _Systems & Control Letters_, vol. 62, no. 3, pp. 286–293, Mar. 2013.
  * [11] R. Gorbet, K. Morris, and D. Wang, “Passivity-based stability and control of hysteresis in smart actuators,” _Control Systems Technology, IEEE Transactions on_ , vol. 9, pp. 5 – 16, 02 2001.
  * [12] O. Waldmann, R. Koch, S. Schromm, P. Müller, I. Bernt, and R. W. Saalfrank, “Butterfly Hysteresis Loop at Nonzero Bias Field in Antiferromagnetic Molecular Rings: Cooling by Adiabatic Magnetization,” _Physical Review Letters_ , vol. 89, no. 24, p. 246401, Nov. 2002.
  * [13] H. Sahota, “Simulation of butterfly loops in ferroelectric materials,” _Continuum Mechanics and Thermodynamics_ , vol. 16, no. 1-2, pp. 163–175, 2004.
  * [14] K. Linnemann, S. Klinkel, and W. Wagner, “A constitutive model for magnetostrictive and piezoelectric materials,” _International Journal of Solids and Structures_ , vol. 46, no. 5, pp. 1149–1166, Mar. 2009. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S002076830800440X
  * [15] J. Linhart, P. Hana, L. Burianova, and S. J. Zhang, “Hysteresis measurements of lead-free ferroelectric ceramics BNBK79 (Bi0.5na0.5) TiO3-(Bi0.5k0.5) TiO3-BaTiO3,” in _2011 10th International Workshop on Electronics, Control, Measurement and Signals_ , Jun. 2011, pp. 1–4.
  * [16] F. Pozo, L. Acho, A. Rodríguez, and G. Pujol, “Nonlinear modeling of hysteretic systems with double hysteretic loops using position and acceleration information,” _Nonlinear Dynamics_ , vol. 57, no. 1-2, p. 1, 2009.
  * [17] F. Pozo and M. Zapateiro, “On the Passivity of Hysteretic Systems with Double Hysteretic Loops,” _Materials_ , vol. 8, no. 12, pp. 8414–8422, Dec. 2015. [Online]. Available: https://www.mdpi.com/1996-1944/8/12/5465
  * [18] B. Drinčić, X. Tan, and D. S. Bernstein, “Why are some hysteresis loops shaped like a butterfly?” _Automatica_ , vol. 47, no. 12, pp. 2658–2664, 2011.
  * [19] B. Jayawardhana, M. A. Vasquez-Beltran, W. J. van de Beek, C. de Jonge, M. Acuautla, S. Damerio, R. Peletier, B. Noheda, and R. Huisman, “Modeling and analysis of butterfly loops via Preisach operators and its application in a piezoelectric material,” in _Proc. IEEE Conf. Decision and Control (CDC)_ , Dec. 2018, pp. 6894–6899.
  * [20] M. A. Vasquez-Beltran, B. Jayawardhana, and R. Peletier, “Asymptotic Stability Analysis of Lur’e Systems With Butterfly Hysteresis Nonlinearities,” _IEEE Control Systems Letters_ , vol. 4, no. 2, pp. 349–354, Apr. 2020.
  * [21] H. Logemann and E. Ryan, “Systems with hysteresis in the feedback loop: existence, regularity and asymptotic behavior of solutions,” _ESAIM: Control, Optimisation and Calculus of Variations_ , vol. 9, pp. 169–196, 2003\.
  * [22] I. D. Mayergoyz, _Mathematical Models of Hysteresis and their Applications_. Academic Press, Oct. 2003, google-Books-ID: AYy6nQjdmIwC.
  * [23] JinHyoung Oh and D. S. Bernstein, “Semilinear Duhem model for rate-independent and rate-dependent hysteresis,” _IEEE Transactions on Automatic Control_ , vol. 50, no. 5, pp. 631–645, May 2005.
  * [24] S. Valadkhan, K. Morris, and A. Khajepour, “Absolute stability analysis for linear systems with Duhem hysteresis operator,” _International Journal of Robust and Nonlinear Control_ , vol. 20, no. 4, pp. 460–471, 2009.
  * [25] A. K. Padthe, J. Oh, and D. S. Bernstein, “Counterclockwise dynamics of a rate-independent semilinear Duhem model,” in _Proceedings of the 44th IEEE Conference on Decision and Control_ , Dec 2005, pp. 8000–8005.
  * [26] C. Visone and W. Zamboni, “Loop Orientation and Preisach Modeling in Hysteresis Systems,” _IEEE Transactions on Magnetics_ , vol. 51, no. 11, pp. 1–4, Nov. 2015.