A Survey of the Hysteretic Duhem Model

Citing a reference allows the author of a scientific article to attribute work and ideas to the correct source. Nonetheless, the process of describing that work and these ideas assumes some interpretation, at least of their relative importance. In order to ensure that the interpretation is reliable, we use, whenever adequate, a quotation from the reference so that the reader has a direct access to the cited source. This direct access is even more important when the source is not easily available like Ref. [67]¹ or is not written in English like Refs. [16, 17, 18, 19, 20, 21, 22] among others, in which case we provide a translation. This is our approach to the literature review of Sect. 2.

In Sects. 4–9 we proceed differently since our aim in these sections is to provide an accurate description of the results presented in the references under study. Because of the diversity of notations and nomenclature in these references, quotations may not be the best way to transmit that accurate description. Instead, we summarize the references using a unifying framework provided in Ref. [35]. The references we have chosen in Sects. 4–9 are, in our opinion, those that are relevant to the subject of this study.

Our aim in this work is also to shed light on the relationships between the concepts introduced in this paper. To this end, we use a special form of the Duhem model, the scalar semilinear one, as a case study.

A brief history of the Duhem model The term hysteresis was coined by J. A. Ewing in 1881 to describe a specific relationship between the torsion of a magnetized wire and its polarization (although the phenomenon of hysteresis has been known and described by several authors before that date as shown in the literature review provided in Ref. [65]).

Quoting from Ewing’s paper [28]: “These curves exhibit, in a striking manner, a persistence of previous state, such as might be caused by molecular friction. The curves for the back and forth twists are irreversible, and include a wide area between them. The change of polarization lags behind the change of torsion. To this action $\ldots$ the author now gives the name Hysterēsis ($\ldots$ to be behind)”.

However, the first in-depth study of hysteresis is due to Pierre Duhem² in the period 1896–1902. A detailed review of Duhem’s work on hysteresis may be found in [67, Chapter IV] so that we provide here only those elements of that extensive work that are directly related to the present paper.

To understand the motivation for Duhem’s work we quote from [67, p. 306]: “take a metallic wire under strain by means of a load. We can take the length of the wire and its temperature as variables that define its state. The gravity weight P will represent the external action. At temperature T and under the load P the wire may be at equilibrium with length l. Give P infinitely small variations, the length l and temperature T will also experience infinitely small variations, and a new equilibrium may be achieved. In this last state, give the gravity weight and temperature variations equal in absolute value, but of opposite signs to the previous ones. The length l should experience a variation equal to the previous one with opposite sign. However, experimentation shows that this is not the case. In general, to the expansion of the wire corresponds a smaller contraction, and the difference lasts with time.”

This permanent deformation is the subject of a seven-memoirs research by Duhem, see Refs. [16]–[22]. In his first memoir submitted to the section of sciences of the Académie de Belgique on October 13, 1894, and reviewed by the mathematician Charles Lagrange in Ref. [44],³ Duhem writes: “The attempts to make the different kinds of permanent deformations compatible with the principles of thermodynamics have been few up till now. Only one of these attempts, due to M. Marcel Brillouin, appears to us worthy of interest.” [16, p. 3]. Duhem analyzes the work of Brillouin and concludes that it is not compatible with the principles of thermodynamics [16, p. 6] (see also [19, pp. 5–7]).

To the best of our knowledge, the first reference that called the form (5) “Duhem model” is Ref. [48] in 1993.⁴ Indeed, the authors of Ref. [43] attributed erroneously Duhem’s model to Madelung [63, p. 797].⁵

Between 1916 when P. Duhem dies and 1993 when his model of hysteresis is finally attributed to him, Duhem’s work on hysteresis does not have a relevant impact. Major references on hysteresis like Refs. [8, 12] or [56] do not cite his memoirs. Several authors propose different forms of the Duhem model without a direct reference to Duhem’s memoirs. This is the case of the Coleman and Hodgdon model of magnetic hysteresis [12], the Dahl model of friction [13], the model (5) in Ref. [3], and a generalized form of the model (5) in Ref. [43, p. 95]. In 1952, Everett cites briefly Duhem’s work as follows [24, p. 751]: “From a thermodynamic standpoint the introduction of an additional variable whose value depends on the history of the system is sufficient for a formal discussion (cf. Duhem $^{\text {[ref]}}$). To advance our understanding of the phenomenon [of hysteresis], however, a molecular interpretation is desirable.”

A general theory of physics based on a molecular interpretation was precisely what Duhem rejected. In a review of his work presented in 1913 for his application to the Académie des Sciences, Duhem writes that his “doctrine should note imitate the numerous mechanical theories proposed by physicists hitherto; to the observable properties that apparatus measure, it will not substitute hidden movements of hypothetical bodies” [67, p. 74].

In recent times, Duhem’s phenomenological approach is becoming more accepted [5, 9, 46, 52, 57]. Indeed, “hysteretic phenomena arising in structural and mechanical systems are so complicated that there has been no well-accepted mathematical model which can describe all observed hysteretic characteristics.” [52, p. 1408]. Moreover, the Preisach model which was believed to describe the constitutive behavior of magnetic hysteresis, has shown to be a phenomenological model [49, p. 2].

Several reasons are invoked for the use of Duhem’s model to describe hysteresis. On the one hand, “differential equation-based models lead to a particularly simple phenomenological description” [46, p. C8–545]. On the other hand, the “Duhem models [sic] $\ldots$ have the advantage that [they] require a small amount of memory so they are suitable in practical and low cost applications.” [9, p. 628]. Finally, many phenomenological models of friction or hysteresis can be seen as particular cases of a more general form of the Duhem model: this is the case for example of the Dahl [13], the LuGre [2, 11], or the Maxwell-slip models [30]. Thus “recast[ing] each model in the form of a generalized $\ldots$ Duhem model $\ldots$ provide[s] a unified framework for comparing the hysteretic nature of these models.”s [57, p. 91].

There are several generalizations of the original Duhem model (5). The following generalization is proposed in [43, p. 95]: $\dot{x}(t)=f\big ( t,x(t),u(t),\dot{u}(t)\big )$. In [64, p. 141] the terms $\phi _{\ell }\big (x(t),u(t)\big )$ and $\phi _{r}\big (x(t),u(t)\big )$ in (5) are replaced by $\left[ \phi _{\ell }\big (x,u\big )\right] (t)$ and $\left[ \phi _{r}\big (x,u\big )\right] (t)$ respectively, where $\phi _{\ell }$ and $\phi _{r}$ are causal operators. In Ref. [54] Duhem’s model is generalized as $\dot{x}(t) = f\big (x(t),u(t) \big )g\big (\dot{u}(t) \big )$ whilst [64, p. 145] proposes the following form for vector hysteresis: $\dot{x}(t) = f\big (x(t),u(t),\pi (\dot{u}) \big )|\dot{u}(t)|$ where $\pi (\dot{u}\ne 0)=\dot{u}/|\dot{u}|$.

Why are there different generalized forms of the Duhem model ? To answer this question we have to recall the concept of rate independence.⁶

To the best of our knowledge, the earliest author to state clearly rate independence is R. Bouc in Ref. [8], although that property was known before Bouc’s work. Due to the importance of rate independence in the study of hysteresis, and the fact that Ref. [8] is not available in English, we quote from [8, p. 17]:“Consider the graph with hysteresis of Fig. 1 where $\mathcal {F}$is not a function of x. To the value$x=x_0$correspond four values of$\mathcal {F}$.

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$\ldots$ If we consider now x as a function of time, the value of the force at instant t will depend not only on the value x(t), but also on all past values of function x since the origin instant where it is defined. If $\beta$ is that instant ($x(\beta )=\mathcal {F}(\beta )=0,\beta \ge -\infty$), then we denote $\mathcal {F}(t)=\mathcal {A}\big (x(\cdot ),t\big )$ the value of the force at instant “$\,t$”, where $x(\cdot )$ “represents” the whole function on the interval $[\beta ,t]^{\text {[footnote]}}$. Our aim is to explicit functional $\mathcal {A}\big (x(\cdot ),t\big )$.

Rate independence is used by Visintin to define hysteresis :“Definition. Hysteresis = Rate Independent Memory Effect.”[64, p. 13]. However, “this definition excludes any viscous-type memory” [64, p. 13] because it leads to rate-dependent effects that increase with velocity. A definition based on rate independence assumes that “the presence of hysteresis loops is not … an essential feature of hysteresis.” [64, p. 14].

This point of view is challenged by Oh and Bernstein who consider hysteresis as a “nontrivial quasi-dc input-output closed curve” [54, p. 631] and propose a modified version of the Duhem model which can represent rate-dependent or rate-independent effects. A characterization of hysteresis systems using hysteresis loops is also addressed by Ikhouane in Ref. [35] through the concepts of consistency and strong consistency.

In light of what has been said it becomes clear that, in Ref. [64], the generalizations of Duhem’s model are done in such a way that rate independence is preserved, whilst a definition of hysteresis based on hysteresis loops in Ref. [54] is compatible with a generalized form of the Duhem model that may be rate dependent or rate independent.

Why are there different models of hysteresis? In Ref. [16] Duhem proposes his model to account for the irreversibility in the modifications of equilibria observed experimentally in magnetic hysteresis [16, Chap. IV], sulfur [17], red phosphorus [19, Chap. III], and in different processes of metallurgy [19].

Preisach [56] uses “plausible hypotheses concerning the physical mechanisms of magnetization” [49, p. 1] to elaborate a model of magnetic hysteresis. This model is also proposed and studied by Everett and co-workers [24, 25, 26, 27] who postulate “that hysteresis is to be attributed in general to the existence in a system of a very large number of independent domains, at least some of which can exhibit metastability.” [24, p. 753].

Krasnosel’skiǐ and Pokrovskiǐ point out to the issue of admissible inputs, as “it is by no means clear a priori for any concrete transducer with hysteresis, how to choose the relevant classes of admissible inputs” [43, p. 5]. This is why they introduce the concept of vibro-correctness which allows the determination of the output of a hysteresis transducer that corresponds to any continuous input, once we know the outputs that correspond to piecewise monotone continuous inputs [43, p. 6]. The models that Krasnosel’skiǐ and Pokrovskiǐ propose (ordinary play, generalized play, hysteron) are vibro-correct, although the authors acknowledge the existence of hysteresis models that may not be vibro-correct like the Duhem model.⁷

Hysteresis models based on a feedback interconnection between a linear system and a static nonlinearity are proposed in Ref. [55]. The authors study “hysteresis arising from a continuum of equilibria … and hysteresis arising from isolated equilibria” [55, p 101].

A review of hysteresis models is provided in Ref. [48] and a detailed study of these (and other) models may be found in Refs. [7, 10, 14, 37], [49, 64].

In light of what has been said, the diversity of hysteresis models is due to the wide range of areas to which hysteresis is concomitant, and the diversity of methods and assumptions underlying the elaboration of these models.

Note that all mathematical models of hysteresis share a common property: they model hysteresis. This fact leads us to our next question.

What is hysteresis? A description found in many papers is that hysteresis “refers to the systems that have memory, where the effects of input to the system are experienced with a certain delay in time.” [33, p. 210]. This description is misleading as it applies also to dynamic linear systems. Indeed, when the output y is related to the input u by $\dot{x}=Ax+Bu$ and $y=Cx$ which is a possible description of a linear system, the output is given by $y(t)=C\big [ \exp (tA)x_0+\int _0^t \exp \big ((t-\tau )A\big )Bu(\tau )\text {d}\tau \big ]$ where $x_0$ is the initial state and $t\ge 0$ is time. We can see that y(t) depends on the integral of a function that incorporates $u(\tau )$ for all $\tau \in [0,t]$, which means that the linear system does have memory. However, “hysteresis is a genuinely nonlinear phenomenon” [10, p. 7].

Mayergoyz considers hysteresis as a rate-independent phenomenon which is “consistent with existing experimental facts.” [49, p. 16]. However, “for very fast input variations, time effects become important and the given definition of rate-independent hysteresis fails.” [49, p. 16]. Also, “in the existing literature, hysteresis phenomenon is by and large linked with the formation of hysteresis loops (looping). This may be misleading and create the impression that looping is the essence of hysteresis. In this respect, the given definition of hysteresis emphasizes the fact that history dependent branching constitutes the essence of hysteresis, while looping is a particular case of branching.” [49, pp. 16–18].

Following Mayergoyz, “All rate-independent hysteresis nonlinearities fall into two general classifications: (a) hysteresis nonlinearities with local memories, and (b) hysteresis nonlinearities with nonlocal memories.” [49, pp. 17]. In a hysteresis with a local memory, the state or output at time $t \ge t_0$ is completely defined by the state or output at instant $t_0$, and the input on $[t_0,t]$. This is the case for example of a hysteresis given by a differential equation. Hysteresis with a nonlocal memory is a hysteresis which is not with local memory. This is the case for example of the Preisach model. “However, the notion of hysteresis nonlinearities with local memories is not consistent with experimental facts.” [49, pp. 19–20]. Hodgdon, on the other hand, writes in relation with the use of a special case of the Duhem model to represent ferromagnetic hysteresis: “These results are in good agreement with the manufacturer’s dc hysteresis data and with experiments” [34, p. 220].

In Ref. [54], Oh and Bernstein consider the generalized Duhem model $\dot{x}=f(x,u)g(\dot{u})$ and $y=h(x,u)$ with u the input, y the output and x the state. The authors assume the existence of a unique solution of the differential equation on the time interval $[0,\infty [$. They also assume the existence of a T–periodic solution $x_T$ for any T–periodic input $u_T$ with one increasing part and one decreasing part, which means that the graph $\{(u_T,x_T)\}$ is a closed curve. Finally they assume that when $T \rightarrow \infty$ the graph $\{(u_T,x_T)\}$ converges with respect to the Hausdorff metric to a closed curve $\mathcal {C}$. If we can find $(a,b_1) \in \mathcal {C}$ and $(a,b_2) \in \mathcal {C}$ with $b_1 \ne b_2$, the curve $\mathcal {C}$ is not trivial and the generalized Duhem model is a hysteresis.

In a PhD thesis advised by Bernstein [15], Drinčić considers systems of the form $\dot{x}=f(x,u)$ and $y=h(x,u)$ for which hysteresis is defined as in Ref. [54]. The system is supposed to be step convergent, that is $\lim _{t \rightarrow \infty } x(t)$ exists for all initial conditions and for all constant inputs. It is noted that there exists “a close relationship” [15, p. 6] between the curve $\mathcal {C}$ and the input-output equilibria map, that is the set $\mathcal {E}=\big \{\big (u,h(\lim _{t \rightarrow \infty } x(t),u)\big )\big \}$ where u is constant and $f\big ( \lim _{t \rightarrow \infty } x(t),u\big )=0$. In particular, the “system $\ldots$is hysteretic if the multivalued map$\mathcal {E}$has either a continuum of equilibria or a bifurcation” [15, p. 7].

In Ref. [6] Bernstein states that “a hysteretic system must be multistable; conversely, a multistable system is hysteretic if increasing and decreasing input signals cause the state to be attracted to different equilibria that give rise to different outputs.” Multistability means that “the system must have multiple attracting equilibria for a constant input value” [6].

In Ref. [50], Morris presents six examples of hysteresis systems taken from the areas of electronics, biology, mechanics, and magnetics; hysteresis being understood as a “characteristic looping behavior of the input-output graph” [50, p. 1]. The author explains the qualitative behavior of these systems from the point of view of multistability. For “the differential equations used to model the Schmitt trigger, cellular signaling and a beam in a magnetic field” it is observed that “these systems, all possess, for a range of constant inputs, several stable equilibrium points.” [50, p. 13]. The author observes that the systems are rate dependent for high input rates.

For the play operator, the Preisach model and the Bouc-Wen model which are rate independent, “these models present a continuum of equilibrium points.” [50, p. 13]. These observations lead the author to conclude that “hysteresis is a phenomenon displayed by forced dynamical systems that have several equilibrium points; along with a time scale for the dynamics that is considerably faster than the time scale on which inputs vary.” [50, p. 13]. Morris proposes the following definition.

“A hysteretic system is one which has (1) multiple stable equilibrium points and (2) dynamics that are considerably faster than the time scale at which inputs are varied.” [50, p. 13].

In Ref. [35], Ikhouane considers a hysteresis operator “$\mathcal {H}$that associates to an input u and initial condition$\xi ^0$ an output $y=\mathcal {H}(u,\xi ^0)$, all belonging to some appropriate spaces.” [35, p. 293]. It is assumed that the operator $\mathcal {H}$ is causal and satisfies the property that constant inputs lead to constant outputs. Examples include all rate-independent models [47, Proposition 2.1], some rate-dependent models, models with local memory like the various generalizations of the Duhem model, and models with nonlocal memory like the Preisach model.

The author introduces two changes in time scale: (1) a linear one which is applied to a given input, and (2) a -possibly- nonlinear one which is the total variation of the original input. When the input is composed with the linear time-scale change, both the input and the output are re-scaled with respect to the total variation of the input, which provides a normalized input independent of the linear time-scale change, and a normalized output. Consistency is defined as being the convergence of the normalized outputs in the space $L^\infty$ endowed with the uniform convergence norm. It is shown that consistency implies the convergence to some set of the graphs output versus input of the hysteresis operator when the linear time scale varies [35, Lemma 9].

Strong consistency is defined as the property that the limit of the normalized outputs, seen a parametrized curve, converges to a periodic orbit which characterizes the hysteresis loop.

The author does not propose a definition of hysteresis, but considers that consistency and strong consistency are properties of a class of hysteresis systems.

Aim of the paper The aim of the paper is to survey the research carried out on the Duhem model from the perspective of its hysteretic properties.

Organization of the paper Section 4 presents some results obtained in Ref. [43], namely the concept of vibro-correctness, sufficient conditions to ensure global solutions of the scalar rate independent Duhem model, and a study of the continuity of the model seen as an operator. Section 5 presents a definition of hysteresis proposed in Ref. [54] that uses a generalized form of Duhem’s model as a tool to get that formal definition. Section 6 presents the concepts of consistency and strong consistency introduced in Ref. [35]. The tools and notations of Ref. [35] are also used as a unifying framework to present the results of the present paper. Section 7 presents a characterization of the generalized Duhem model obtained in Ref. [51]. Section 8 summarizes the results obtained in Ref. [40] in relation with the study of the dissipativity of the Duhem model. Section 9 summarizes some results obtained in Ref. [64] in relation with the existence of a Duhem operator, its smoothness, and some generalizations of the model. Section 10 is a note that explores the minor loops of hysteresis systems with particular emphasis on the Duhem model. For ease of reference, some results on the existence and uniqueness of the solutions of differential equations are presented in Appendix 15.

To illustrate the results obtained in Sects. 4–10, and to analyze the relationships between these results, we use the scalar semilinear Duhem model as a case study. The corresponding mathematical analysis is stated in various lemmas and theorems provided in Section 11, whose proofs are given in 16–20. The relationships between the results obtained in Sects. 4–9 are commented upon in Section 12. These comments lead to the formulation of several open problems in Sect. 13 and a conjecture in Sect. 11.9.

A real number x is said to be strictly positive when $x>0$, strictly negative when $x<0$, nonpositive when $x \le 0$, and nonnegative when $x \ge 0$. A function $h:\mathbb {R} \rightarrow \mathbb {R}$ is said to be strictly increasing when $t_1<t_2 \Rightarrow h(t_1) <h(t_2)$, strictly decreasing when $t_1<t_2 \Rightarrow h(t_1)>h(t_2)$, nonincreasing when $t_1<t_2 \Rightarrow h(t_1) \ge h(t_2)$, and nondecreasing when $t_1<t_2 \Rightarrow h(t_1) \le h(t_2)$.⁸

An ordered pair a, b is denoted (a, b) whilst the open interval $\{t\in \mathbb {R}\mid a<t<b\}$ is denoted ]a, b[. The set of nonnegative integers is denoted $\mathbb {N}=\{0,1,\ldots \}$ and the set of nonnegative real numbers is denoted $\mathbb {R}_+=[0,\infty [$.

The Lebesgue measure on $\mathbb {R}$ is denoted $\mu$. We say that a subset of $\mathbb {R}$ is measurable when it is Lebesgue measurable. Let $I \subset \mathbb {R}_+$ be an interval, and consider a function $\phi :I \rightarrow \mathbb {R}^l$ where $l>0$ is an integer. We say that $\phi$ is measurable when $\phi$ is $(M_\mu ,B)$–measurable where B is the class of Borel sets of $\mathbb {R}^l$ and $M_\mu$ is the class of measurable sets of $\mathbb {R}_+$ [66]. For a measurable function $\phi :I \rightarrow \mathbb {R}^l$, $\Vert \phi \Vert _{I}$ denotes the essential supremum of the function $|\phi |$ on I where $|\cdot |$ is the Euclidean norm on $\mathbb {R}^l$. When $I=\mathbb {R}_+$, this essential supremum is denoted $\Vert \phi \Vert$.

$W^{1,\infty }(\mathbb {R}_+,\mathbb {R}^l)$ denotes the Sobolev space of absolutely continuous functions $\phi :\mathbb {R}_+ \rightarrow \mathbb {R}^l$. For this class of functions, we have $\Vert \phi \Vert <\infty$; the derivative of $\phi$ is denoted $\dot{\phi }$; this derivative is defined almost everywhere and satisfies $\Vert \dot{\phi }\Vert <\infty$. Endowed with the norm $\Vert \phi \Vert _{W^{1,\infty }(\mathbb {R}_+,\mathbb {R}^l)}=\max \left( \Vert \phi \Vert , \Vert \dot{\phi }\Vert \right)$, the vector space $W^{1,\infty }(\mathbb {R}_+,\mathbb {R}^l)$ is a Banach space [45, pp. 280–281].

$L^{\infty }(\mathbb {R}_+,\mathbb {R}^l)$ denotes the Banach space of measurable and essentially bounded functions $\phi :\mathbb {R}_+ \rightarrow \mathbb {R}^l$ endowed with the norm $\Vert \cdot \Vert$.

$C^{0}(\mathbb {R}_+,\mathbb {R}^l)$ denotes the Banach space of continuous functions $\phi :\mathbb {R}_+ \rightarrow \mathbb {R}^l$ endowed with the norm $\Vert \cdot \Vert$.

$\forall \gamma \in \;]0,\infty [$, the linear time-scale-change $s_{\gamma }:\mathbb {R_+} \rightarrow \mathbb {R_+}$ is defined by the relation $s_{\gamma }(t)=t/\gamma , \forall t \in \mathbb {R}_+$.

$\displaystyle {\lim _{x \uparrow a}}$ sets for $\displaystyle {\lim _{\begin{array}{c} x \rightarrow a \\ x <a \end{array}}}$ whilst $\displaystyle {\lim _{x \downarrow a}}$ sets for $\displaystyle {\lim _{\begin{array}{c} x \rightarrow a \\ x>a \end{array}}}$.

Let U be a set and let $T \in \;]0,\infty [$. The function $\phi :\mathbb {R}_+ \rightarrow U$ is said to be T–periodic if $\phi (t)=\phi (t+T), \forall t \in \mathbb {R}_+$.

This section presents those results obtained in Ref. [43] that are relevant to the present paper. This is in particular the case of the concept of vibro-correctness which allows to extend the set of admissible inputs from continuously differentiable to continuous.

This section presents those results obtained in Ref. [54] that are relevant to the present paper. In particular, the authors of Ref. [54] propose a definition that decides whether a given generalized Duhem model is a hysteresis or not.

This section presents those results obtained in Ref. [35] that are relevant to the present paper. This is in particuler the case for the concepts of consistency and strong consistency.

This section presents those results obtained in Ref. [51] that are relevant to the present paper. In particular, Ref. [51] characterizes the function g that appears in Eq. (17).

Consider the generalized Duhem model (17)–(18) under Assumption 1. Let $\lambda \in \; ]0,\infty [$.

Assumption 5 implies that $\lambda$ is unique, and the function g is said to be of class $\lambda$.

Under Assumptions 1, 5, and 6 we have the following.

Lemma 6 says that if $\lambda \ne 1$ then the corresponding generalized Duhem model does not represent a hysteresis behavior.¹⁷ Thus, the existence of $\lim _{w\downarrow 0}\frac{g\left( w\right) }{w}$ and $\lim _{w\uparrow 0}\frac{g\left( w\right) }{|w|}$ is a necessary condition for the generalized Duhem model to represent a hysteresis. This necessary condition has been derived from the concept of consistency presented in Sect. 6.4. Note that this condition has been assumed for the semilinear Duhem model proposed in Ref. [54] (see Eq. (68) along with Eqs. (66)–(67)).

This section presents those results obtained in Ref. [40] that are relevant to the present paper. This is the case for the dissipativity of a special form of the Duhem model. The concept of dissipativity/passivity is treated in [42, chapter 6] as an abstracted form of energy dissipation which makes this concept relevant to the study of hysteresis.

This section presents those results obtained in Ref. [64] that are relevant to the present paper. In particular, a local Lipschitz property of the Duhem model is provided.

It is shown in [64, Theorem 1.5] that the operator $\mathcal {M}$ can be extended to an operator $\bar{\mathcal {M}}:C^0\big ([0,\mathcal {T}],\mathbb {R} \big ) \cap BV\big ([0,\mathcal {T}],\mathbb {R} \big ) \times \mathbb {R} \rightarrow C^0\big ([0,\mathcal {T}],\mathbb {R} \big ) \cap BV\big ([0,\mathcal {T}],\mathbb {R} \big )$ where BV is the space of functions that have bounded total variation.

Section 12.2 provides comments on the relationship between Proposition 2 and the effect of noise on the hysteresis loop of the Duhem model.

The minor loops of the Duhem model have not been studied formally in the available literature. However, their behavior is important as evidenced by the large number of published works dedicated to their study both from an experimental point of view, and from a mathematical point a view for the Preisach model (see for example Ref. [49] and the references therein).

For this reason, we provide in this section the formal definition of a minor loop and analyze the behavior of the minor loops of the scalar semilinear Duhem model in Sect. 11.9. The material provided in this section may be used as a platform to attract mathematicians to the formal analysis of the minor loops of the Duhem model.

In magnetic hysteresis, when magnetization M is plotted against magnetic field H the following is observed. The curve $\big (H(t),M(t)\big )$ follows the path $P_1 \rightarrow P_2$ when H increases with time t (see Fig. 2). Then the path $P_2 \rightarrow P_3$ is followed when H decreases. What is important to note is that, when H increases again from the point $P_3$, the path followed by $\big (H(t),M(t)\big )$ ends precisely at the point$P_2$ (see for example Ref. [31]).

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The loop formed by the path $P_2 \rightarrow P_3 \rightarrow P_2$ is called a minor loop. It occurs in electromagnetic devices when the input is periodic but not exactly sinusoidal. The distortion of the input generates minor loops when hysteresis is involved which causes energy losses. This fact explains the interest of analyzing the behavior of minor loops.

In what follows we formalize mathematically the behavior observed in Fig. 2.

For $u \in \mathbb {M}_{u_{\min ,1},u_{\min ,2},u_{\max ,1},u_{\max ,2},\alpha _1,\alpha _2,\alpha _3,T}$ define $\varrho _i=\rho _u(\alpha _i),\; i=1,2,3$. Then we have

$$\begin{aligned} \varrho _1=&\;u_{\max ,1}-u_{\min ,1}, \end{aligned}$$

(53)

$$\begin{aligned} \varrho _2=&\;\varrho _1+u_{\max ,1}-u_{\min ,2}, \end{aligned}$$

(54)

$$\begin{aligned} \varrho _3=&\;\varrho _2+u_{\max ,2}-u_{\min ,2}, \end{aligned}$$

(55)

$$\begin{aligned} \rho _u(T)=&\;\varrho _4=\varrho _3+u_{\max ,2}-u_{\min ,1}. \end{aligned}$$

(56)

The function $\psi _u \in W^{1,\infty }(\mathbb {R}_+,\mathbb {R})$ in $\varrho _4$–periodic by Lemma 5, and can be determined using Lemma 1 as

$$\begin{aligned} \psi _u(\varrho )=&\;\varrho +u_{\min ,1},\forall \varrho \in [0,\varrho _1], \end{aligned}$$

(57)

$$\begin{aligned} \psi _u(\varrho )=&\;-\varrho +\varrho _1+u_{\max ,1},\forall \varrho \in [\varrho _1,\varrho _2], \end{aligned}$$

(58)

$$\begin{aligned} \psi _u(\varrho )=&\;\varrho -\varrho _2+u_{\min ,2},\forall \varrho \in [\varrho _2,\varrho _3], \end{aligned}$$

(59)

$$\begin{aligned} \psi _u(\varrho )=&\;-\varrho +\varrho _3+u_{\max ,2},\forall \varrho \in [\varrho _3,\varrho _4]. \end{aligned}$$

(60)

Define

$$\begin{aligned} \varrho _5=&\;u_{\max ,1}-u_{\min ,2}+\varrho _2 \in \;]\varrho _2,\varrho _3], \\ \varrho _6=&\;u_{\min ,2}-u_{\min ,1} \in \;]0,\varrho _1[,\\ \varrho _7=&\;\varrho _3 +u_{\max ,2}-u_{\min ,2} \in \;]\varrho _3,\varrho _4[. \end{aligned}$$

Then $\psi _u(\varrho _1)=\psi _u(\varrho _5)=u_{\max ,1}$ and $\psi _u(\varrho _2)=\psi _u(\varrho _6)=\psi (\varrho _7)=u_{\min ,2}$. Figure 3 illustrates what has been exposed up till now.

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Note that the hysteresis loop $\mathcal {G}_u$ of Equation (31) is such that $\mathcal {G}_u=\mathcal {V}_u \cup \mathcal {N}_u$. Figure 4 provides an example of a minor loop and a major loop that correspond to the normalized input of Figure 3.

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The concepts introduced in this section are applied to the scalar semilinear Duhem model in Sect. 11.9.

In this section we use the semilinear Duhem model to illustrate the concepts presented in this paper, and to analyze the relationships between these concepts. Section 11.1 presents the model. In Sect. 11.2 we provide sufficient conditions for the consistency of the model. Section 11.3 focuses on the study of the strong consistency of the semilinear Duhem model. The results of Sects. 11.2 and 11.3 are illustrated by numerical simulations in Sect. 11.4. In Sect. 11.5 we specialize into the scalar version of the semilinear Duhem model. Section 11.5 provides the conditions under which the scalar semilinear Duhem model is a hysteresis according to Definition 4. The results of Sect. 11.5 are illustrated by numerical simuations in Sect. 11.6. The relationship between Definition 4 and strong consistency is commented upon in Sect. 12.1. Section 11.7 analyzes the dissipativity of the scalar rate-independent semilinear Duhem model. The results of Sect. 11.7 are illustrated by numerical simulations in Sect. 11.8. The relationship between dissipativity and orientation of the hysteresis loop is commented upon in Sect. 12.3. The minor loops of the scalar semilinear Duhem model are studied and commented upon in Sect. 11.9.

11.4 Illustration of the Consistency and Strong Consistency of the Scalar Semilinear Duhem Model

We consider the semilinear Duhem model with the following parameters: $n=1,$$A_1=-1,$$A_2=1,$$B_1=1,$$B_2=-1,$$E_1=0,$$E_2=0,$$C=1,$$D=0.$ The function $g_1:\mathbb {R} \rightarrow \mathbb {R}$ is defined by the relations $\forall x \in \mathbb {R}, g_1(x)=0$ if $x \le 0$, and $g_1(x)=x+x^2$ if $x \ge 0$. The function $g_2:\mathbb {R} \rightarrow \mathbb {R}$ is defined by the relations $\forall x \in \mathbb {R}, g_2(x)=0$ if $x \ge 0$, and $g_2(x)=x$ if $x \le 0$.

Consider the 2–periodic input u defined as follows: $u(t)=t, \forall t \in [0,1]$, and $u(t)=2-t,\forall t \in [1,2]$ (see Fig. 5).

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Observe that, since $\rho _u$ is the identity function, we have $v_u=\dot{u}$ almost everywhere so that in the differential equation (73) we have $v_u(\varrho )=1,\forall \varrho \in \; ]0,1[$ and $v_u(\varrho )=-1,\forall \varrho \in \; ]1,2[$. The following values of $\gamma$ are considered: $\gamma =1$, $\gamma =10$ and $\gamma =100$. The differential equation (73) is solved using Matlab solver ode23s for the three values of $\gamma$ and with the initial condition $x_0=0$. For each value of $\gamma$ we obtain the corresponding $x_{u \circ \gamma }$ which, in this case, is equal to $\varphi _{u \circ \gamma }$ as $C=1$ and $D=0$ [see Eq. (72)]. Figure 6 provides the plot of function $\varphi _{u \circ \gamma }(\varrho )$ versus time $\varrho$ for $\gamma =1$, $\gamma =10$ and $\gamma =100$ (dotted). The same figure provides the plot of function $\varphi _{u}^\star (\varrho )$ versus time $\varrho$ (solid). The plot of $\varphi _{u}^\star$ has been obtained by solving the differential equation (74) using Matlab solver ode23s, and taking into account that $\psi _u=u$ and that the initial condition $\varphi _{u}^\star (0)$ is also $x_0=0$ [see Eq. (75)]. Since $C=1$ and $D=0$ we have $\varphi _{u}^\star =x_u^\star$ [see Eq. (76)]. We can see that the plots $\varphi _{u \circ \gamma }(\varrho )$ versus $\varrho$ converge to the plot $\varphi _{u}^\star (\varrho )$ versus $\varrho$ as $\gamma$ increases which is predicted by Theorem 6.

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Now that $\varphi _{u}^\star$ has been computed, the functions $\varphi _{u,k}^{\star }$ where $k\in \mathbb {N}$ are determined using Eq. (78). Figure 7 provides the plots of function $\varphi _{u,k}^{\star }(\varrho )$ versus $\varrho$ for $k=0$, $k=1$ and $k=2$ (dotted). The same figure provides the plot of the function $\varphi _{u}^\circ (\varrho )$ versus time $\varrho$ (solid). The plot of $\varphi _{u}^\circ$ is obtained by solving the differential equation (79) using Matlab solver ode23s, and taking into account that $\psi _u=u$. The initial condition $x_u^\circ (0)$ is obtained from Eq. (90). Note that we have $\varphi _{u}^\circ =x_u^\circ$ as $C=1$ and $D=0$ (see Eq. (80)). As predicted by Theorem 7 it can be seen that the plots $\varphi _{u,k}^{\star }(\varrho )$ versus $\varrho$ converge to the plot $\varphi _{u}^\circ (\varrho )$ versus $\varrho$ as k increases.

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The hysteresis loop of the operator $\mathcal {H}_o$ with respect to $(u,x_0)$, that is the set $\left\{ \big (\psi _u(\varrho ),\varphi _{u}^\circ (\varrho )\big ),\varrho \in [0,2]\right\}$ (see Eq. (31)), is plotted in Fig. 8. It can be seen that the hysteresis loop is not trivial as predicted by Lemma 8 since Equality (97) does not hold.

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We now use the value $E_2=2$ instead of $E_2=0$ so that both Equalities (96) and (97) hold. Lemma 8 predicts that the hysteresis loop is trivial as can be observed in Fig. 9.

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11.6 Illustration of the Hysteresis Property-According to Definition 4- of the Semilinear Duhem Model

We consider the same scalar semilinear Duhem model as in Sect. 11.4, that is we consider that

$$\begin{aligned} \dot{x}(t)&=g_1\big (\dot{u}(t)\big )\big (-x(t)+u(t) \big )+g_2\big (\dot{u}(t)\big )\big ( x(t)-u(t)\big )\\&\text { for almost all } t \in \mathbb {R}_+, \\ x(0)&=x_0,\\ y(t)&=x(t), \forall t \in \mathbb {R}_+. \end{aligned}$$

We take as initial condition $x_0=0$, and as input the 2–periodic function u defined as follows: $u(t)=t, \forall t \in [0,1]$, and $u(t)=2-t,\forall t \in [1,2]$ (see Fig. 5). Let $\gamma \in \;]0,\infty [$ and consider the output $\mathcal {H}_o(u \circ s_\gamma ,x_{0})=x_\gamma$ which is the solution of the differential equation (140). We take $\gamma = 1$ and solve (140) using Matlab solver ode23s. The resulting solution is plotted against the input $u \circ s_\gamma$ in Fig. 10 (dotted).

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The value $x_{0,\gamma }$ is computed using Eq. (103); we get $x_{0,\gamma } \simeq 0.4979$. The fact that $x_{0,\gamma }\ne x_0$ explains why the set $\left\{ \big (u\circ s_\gamma (t),[\mathcal {H}_o(u \circ s_\gamma ,x_{0})](t)\big ),t\in \mathbb {R}_+\right\}$ is not a closed curve. We now solve the differential equation (140) taking as initial condition $x(0)=x_{0,\gamma }$. The obtained solution is plotted against the input $u \circ s_\gamma$ in Fig. 10 (solid). We can see that the set $\mathcal {C}_{u,\gamma }=\left\{ \big (u\circ s_\gamma (t),[\mathcal {H}_o(u \circ s_\gamma ,x_{0,\gamma })](t)\big ),t\in \mathbb {R}_+\right\}$ is a curve which is closed as predicted by Theorem 8.

In Fig. 10 observe that the point $\big (u\circ s_\gamma (t),[\mathcal {H}_o(u \circ s_\gamma ,x_{0})](t)\big )$ gets closer to the closed curve $\mathcal {C}_{u,\gamma }$ as $t \rightarrow \infty$. This is a consequence of the uniform convergence of $z_m$ to $\bar{z}_\gamma$ on the interval [0, T] (see the proof of Theorem 8).

Now we plot the closed curve $\mathcal {C}_{u,\gamma }$ for $\gamma =1$, $\gamma =10$ and $\gamma =100$ (see Fig. 11). The closed curve $\mathcal {C}_u$ is plotted using Eq. (106) and the explicit expressions of the functions $\xi _1$ and $\xi _2$ provided in Eqs. (93)–(94). We observe that $\mathcal {C}_{u,\gamma }$ gets closer to the closed curve $\mathcal {C}_u$ as $\gamma$ increases as predicted by Theorem 9 which shows that Condition (i) of Definition 4 is fulfiled.

Regarding Condition (ii) of Definition 4, observe that Eq. (108) does not hold in our case. Thus, using Lemma 9, it follows that Condition (ii) of Definition 4 holds. This fact can be observed in Fig. 11 since to any input value $\nu \in \;]u_{\min },u_{\max }[\,=\,]0,1[$ correspond two different values $\xi _1(\nu )$ ($\bullet$ marker) and $\xi _2(\nu )$ ($\star$ marker).

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11.8 Illustration of the Dissipativity of the Scalar Rate-Independent Semilinear Duhem Model

Consider the model (109)–(112) with parameters $A_1=-1,$$A_2=1$, $B_1=1$, $B_2=-1$, $E_1=E_2=0$, $C=1$, $D=0.$ With these values the relations (114)–(116) hold. The anhysteresis function is given by $f_{\text {an}}(v)=v$, and it is possible to find the intersecting function $\Omega$ explicitly. We get

$$\begin{aligned} \Omega (x_0,u_0) = {\left\{ \begin{array}{ll} u_0+\log (x_0-u_0+1)&{} \text { if } \quad x_0 \ge u_0,\\ u_0-\log (-x_0+u_0+1) &{} \text { if } \quad x_0 \le u_0, \end{array}\right. } \end{aligned}$$

(118)

where $\log$ sets for the natural logarithm. The function $\omega _{\Phi }$ in (41) is given by

$$\begin{aligned} \omega _{\Phi }(\sigma ,x_1,v) = {\left\{ \begin{array}{ll} \sigma -1+(x_1-v+1)e^{v-\sigma }&{} \text { if } \quad \sigma \ge v,\\ \sigma +1+(x_1-v-1)e^{\sigma -v}&{} \text { if } \quad \sigma \le v, \end{array}\right. } \end{aligned}$$

(119)

and the function $\varsigma$ in (42) is given by

$$\begin{aligned} \varsigma (x_1,v) = {\left\{ \begin{array}{ll} x_1v-v-\log (x_1-v+1)-\frac{v^2}{2}+x_1&{}\\ \text { if } \quad x_1 \ge v,&{}\\ x_1v+v-\log (-x_1+v+1)-\frac{v^2}{2}-x_1&{}\\ \text { if } \quad x_1 \le v.&{} \end{array}\right. } \end{aligned}$$

(120)

We take as initial condition $x_0=0$. Now, consider the 2–periodic input u defined as follows: $u(t)=t, \forall t \in [0,1]$, and $u(t)=2-t,\forall t \in [1,2]$ (see Figure 5). Note that $\forall t \in \mathbb {R}_+,u(t) \in \left[ \frac{1}{A_1},\frac{1}{A_2} \right] =[-1,1]$. The curve x(t) ($=y(t)$) versus u(t) is plotted in Fig. 12. As predicted by Lemma 10 it can be seen that $t \mapsto \big (u(t),x(t)\big )$ is counterclockwise.

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Now take as new input the 2–periodic function u defined as follows: $u(t)=t-3, \forall t \in [0,1]$, and $u(t)=-1-t,\forall t \in [1,2]$ (see Fig. 13). Observe that the input is not in the interval $[-1,1]$.

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The curve $t \mapsto \big (u(t),y(t)\big )$ is provided in Fig. 14. It can be seen that $t \mapsto \big (u(t),y(t)\big )$ is not counterclockwise.

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11.9 Minor Loops of the Scalar Semilinear Duhem Model

In this section we apply the concepts introduced in Sect. 10 to the scalar semilinear Duhem model.

Lemma 11

Consider the semilinear Duhem model (63)–(65) with$n=1$, $A_1<0$, $A_2>0$. If Assumption8 holds, then Equalities (96)–(97) hold, and $\forall (u,x_0) \in \Lambda \times \mathbb {R}$ the operator$\mathcal {H}_{o}$ has a trivial hysteresis loop with respect to$(u,x_0)$ (see Definition9).

Proof

See Appendix 21.

To illustrate Lemma 11 consider the semilinear Duhem model of Sect. 11.4 with $E_2=0$, and the input $u=\psi _u$ given by Eqs. (209)–(212) for $\alpha =0.5$ (see Fig. 15).

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The corresponding hysteresis loop is the set $\big \{\big (\psi _u(\varrho ),\varphi ^\circ _u(\varrho ))\big ),\varrho \in [0,\varrho _4=3]\big \}$ where $\varphi ^\circ _u$ obeys Eqs. (79)–(80), and the initial condition is given by Equation (234). The hysteresis loop is provided in Fig. 16. Observe that $\psi _u(\varrho _1)=\psi _u(\varrho _3=\varrho _5)$ and that $\varphi _{u}^\circ (\varrho _1) \ne \varphi _{u}^\circ (\varrho _3)$. This is due to the fact that Equality (97) does not hold so that Assumption 8 is not valid by Lemma 11.

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We now use the value $E_2=2$ instead of $E_2=0$ so that both equalities (96) and (97) hold, which is a necessary condition for Assumption 8 to hold. We consider the input $u \in \Lambda$ of Fig. 5. The corresponding hysteresis loop is reported in Fig. 9: it is a line. This means that the operator $\mathcal {H}_{o}$ has a trivial hysteresis loop with respect to $(u,x_0)$ as predicted by Lemma 11.

Lemma 11 says that the scalar semilinear Duhem model cannot represent the hysteresis behavior observed in magnetic hysteresis. Indeed, to produce minor loops that satisfy Assumption 8, the hysteresis loop of the model should be trivial.

This observation leads to the following conjecture.

Conjecture 1

Consider the generalized Duhem model (17)–(19). Assume that the corresponding operators $\mathcal {H}_{o}$ and $\mathcal {H}_{s}$ are consistent with respect to all $(u,x_0) \in W^{1,\infty }(\mathbb {R}_+,\mathbb {R}) \times \mathbb {R}^n$ and are strongly consistent with respect to all periodic inputs $u \in W^{1,\infty }(\mathbb {R}_+,\mathbb {R})$ and all initial states $x_0 \in \mathbb {R}^n$. If Assumption 8 holds, then $\forall (u,x_0) \in \Lambda \times \mathbb {R}^n$, the operators $\mathcal {H}_{o}$ and $\mathcal {H}_{s}$ have a trivial hysteresis loop with respect to $(u,x_0)$ (see Definition 9).

If true, the conjecture would mean that the Duhem model -in its generalized form- is not able to describe the minor loops in magnetic hysteresis.

However, in several engineering problems, the Duhem model is not used to reproduce the behavior of minor loops in magnetic hysteresis. For example, in control problems, it is not necessary to have an accurate model that describes the controlled process with precision. Instead, an approximate model may be appropriate if it captures some essential features of the controlled plant, and at the same time, is simple enough to allow the design of a relatively simple controller (see for example Ref. [36]).

In this section we explore the connections that exist between the concepts presented in this paper. We use the case study of the semilinear Duhem model to illustrate these connections and motivate the open problems proposed in Sect. 13.

More research is needed to better understand Duhem’s model seen as a class of differential equations, and also as a representation of hysteresis. In particular, it is important to get answers to the open problems -and to the conjecture- proposed in this paper.