# A Survey of the Hysteretic Duhem Model

Abstract: The Duhem model is a simulacrum of a complex and hazy reality: hysteresis. Introduced by Pierre Duhem to provide a mathematical representation of thermodynamical irreversibility, it is used to describe hysteresis in other areas of science and engineering. Our aim is to survey the relationship between the Duhem model as a mathematical representation, and hysteresis as the object of that representation.

## Summary (3 min read)

### 2 Introduction and literature review

- Give P infinitely small variations, the length l and temperature T will also experience infinitely small variations, and a new equilibrium may be achieved.
- There are several generalizations of the original Duhem model (5).
- Multistability means that “the system must have multiple attracting equilibria for a constant input value” [6].

### 3 Terminology and notations

- This section presents those results obtained in Ref. [43] that are relevant to the present paper.
- Instead, the authors of Ref. [54] propose a definition that decides whether a given generalized Duhem model is a hysteresis or not (this is Definition 4).
- Finally the relationship between Definition 4 and strong consistency is analyzed in Section 12.1. 6 A summary of the results obtained in Ref. [35].
- Lemma 1 shows that the input u has been “normalized” so that the resulting function ψu is such that ψ̇u has norm 1 with respect to the new time variable %.

### 10 A note on minor loops

- The minor loops of the Duhem model have not been studied formally in the available literature.
- For this reason, the authors 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 Section 11.9.
- Figure 3 illustrates what has been exposed up till now.
- Depending on the particular field in which hysteresis is observed, minor loops may have some additional properties that may be formalized mathematically.
- Property (i) says that the major loop and the minor loop intersect at only one point when umax,1 < umax,2. Property (ii) says that the minor loop is located inside the major loop.

### 12 Relationships between concepts

- In this section the authors explore the connections that exist between the concepts presented in this paper.
- The authors use the case study of the semilinear Duhem model to illustrate these connections and motivate the open problems proposed in Section 13.
- 1 Relationship between Definition 4 and strong consistency.

### 12.1.1 Comments on Definition 4

- The authors have seen in Section 5.2 that Ref. [54] proposes a definition that aims to decide whether a given generalized Duhem model is a hysteresis or not.
- (iii) Check whether Condition (i) of Definition 4 holds.
- Indeed, the concept of Cauchy sequence can be used to prove the existence of x0,γ without actually having to find the explicit expression of x0,γ .
- This equality is obtained thanks to the explicit expression (103) of the initial condition x0,γ .
- To sum up, the linearity with respect to the state in the differential equation that describes the scalar semilinear Duhem model, is crucial to prove that Condition (i) of Definition 4 holds.

### 12.1.2 Comments on strong consistency

- The analysis of the consistency of the semilinear Duhem model is provided in Section 11.2, and it uses both the linearity with respect to the state, and the fact that the initial condition in Equation (70) does not change with γ.
- For the generalized Duhem model (17) that may not be linear with respect to the state, Lemma 6 provides sufficient conditions that provide the expression of the corresponding rate independent Duhem model.
- Ensuring these sufficient conditions may not be easy if the model is nonlinear with respect to the state.
- Instead, the concept of Cauchy sequence is used in Ref. [35] to prove the desired convergence property.
- Again, the linearity of the model is used to derive a Lyapunov function which allows mathematical analysis.

### 12.1.3 Relationship between the hysteresis loop derived from Definition 4 and the one derived from strong consistency

- The hysteresis loop derived from strong consistency is the set Gu of Equation (31).
- It can be checked that, for the scalar semilinear Duhem model, the authors have Cu = Gu. However, for the generalized Duhem model, at the time of the submission of the present paper they do not know whether the sets Cu and Gu are equal or not.
- This means that, in order to evaluate the effect of perturbations on the hysteresis loop of the model (43)–(45), Proposition 2 needs to be enhanced to take into account different initial conditions.
- This observation leads to formulating Open Prob- lem 2 in Section 13.2.

### 13 Open problems

- The motivation for Open Problem 1 is provided in Section 12.1.3.
- Consider that the generalized Duhem model (17)– (19) satisfies Assumption 1 so that the authors can define the operators Ho and Hs of Section 5.1.
- The motivation for Open Problem 2 is provided in Section 12.2.
- Consider the scalar rate-independent Duhem model (34)–(36) where the output is the state x.

### A On the existence and uniqueness of solutions of differential equations

- In this section the authors present some existence and uniqueness theorems for the solutions of ordinary differential equations.
- The differential equation (129)–(130) has a solution on some nonempty open interval I 3 t0, in the sense that there exists an absolutely continuous function x : I → Rn such that the following properties (i)–(iii) are satisfied.

### D Proof of Theorem 9

- This is the aim of the following analysis.
- The uniform convergence of z̄γ (restricted to the interval [0, T ]) to z̄ has thus been demonstrated, which completes the proof.

### F Proof of Lemma 10

- It can be checked from Equation (204) that this last equality cannot hold.
- A similar argument can be used for Equation (205).

### G Proof of Lemma 11

- Observe that, for Theorem 5 to hold, it is needed that A1 and −A2 are both stable.
- Using Theorem 7 it follows that the function ϕ◦u that characterizes the hysteresis loop satisfies the differential state equation (79) and the output equation (80).
- This study was funded by the Spanish Ministry of Economy, Industry and Competitiveness (grant number DPI2016-77407-P (AEI/FEDER, UE)).

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