This is a preprint of: “Minor loops of the Dahl and LuGre models”, Faycal Ikhouane, Vı́ctor Mañosa, Gisela Pujol, Appl. Math. Model., vol. 77, 1679–1690, 2020. DOI: [10.1016/j.apm.2019.08.031] Minor loops of the Dahl and LuGre models Fayçal Ikhouanea,∗, Vı́ctor Mañosab , Gisela Pujolb Departament de Matemàtiques, Universitat Politècnica de Catalunya a EEBE, Av. Eduard Maristany 16, 08019 Barcelona, Spain b ESEIAAT, Colom 1 and 11, 08222 Terrassa, Spain Abstract Hysteresis is a special type of behavior encountered in physical systems: in a hysteretic system, when the input is periodic and varies slowly, the steady-state part of the output-versus-input graph becomes a loop called hysteresis loop. In the presence of perturbed inputs, this hysteresis loop presents small lobes called minor loops that are located inside a larger curve called major loop. The study of minor loops is being increasingly popular since it leads to a quantification of the loss of energy. The aim of the present paper is to give an explicit analytic expression of the minor loops of the LuGre and the Dahl models of dynamic dry friction. Keywords: Hysteresis; Minor loops; LuGre and Dahl models. MSC 2010: 34C55; 93A30; 93A99; 46T99; PACS: 77.80.Dj; 75.60.-d. 1. Introduction Hysteresis is a nonlinear phenomenon observed in some physical systems under low-frequency excitations. It appears in many areas as biology, electronics, ferroelas- ticity, magnetism, mechanics or optics [3, 4, 14, 16, 22]. This phenomenon is currently classified into two categories: rate independent (RI) and rate dependent (RD) hys- teresis. For RI hysteresis, the output-versus-input graph of the hysteresis system does not change with the frequency of the input signal. This is the case for example of the Bouc-Wen or the Preisach models, see [13] and [17] respectively. For RD hys- teresis, the output-versus-input graph of the hysteresis system may change with the frequency, but it converges in some sense to a fixed loop called the hysteresis loop ∗ Corresponding author Preprint submitted to Elsevier July 24, 2019 when the frequency goes to zero. This is the case for example of the LuGre model and the semilinear Duhem model, see [19] and [12, 20] respectively. Research in the field of hysteresis has focused mainly on the study of rate-independent hysteresis, and it is only in the last 15 years that the importance of rate-dependent phenomena has been acknowledged, and it constitutes a challenge by itself. The recent years have witnessed a growing interest in a phenomenon that appears in hysteretic systems under perturbed periodic signals: the hysteresis loop shows to be composed of a big cycle called major loop, and one or several small lobes called minor loops located inside the major loop. Figure 1 shows the hysteresis loop of a magnetic system when the input is the one of Figure 2, see [10] for instance. Major loop Minor loop umax,2 umin,1 umin,2 umax,1 umax,2 Figure 1: Hysteresis loop of a magnetic system with the input of Figure 2. Black: major loop. Grey: minor loop. umax,2 umax,1 umin,2 umin,1 t1 t2 t5 t3 t4 = T Figure 2: A bimodal T –periodic input u(t) versus t. This interest in the study of minors loops is due in part to the fact that minor 2 loops are related to a loss of energy, see [24] for instance. From a formal point of view, minor loops have been studied mainly in relation with the Preisach model [17, p.19]. Apart from [12, Sections 10, 11.9] we are not aware of any mathematical analysis of minor loops of hysteresis systems given by differential equations. The aim of the present paper is to fill this void by providing an explicit analytic description of the minor loops of the Dahl and the LuGre models. The Dahl model is an idealization of dynamic dry friction proposed by Dahl in 1976 [6]. This model relates an input displacement u to an output force y as y(t) = Fc w(t),  ẇ(t) = ρ u̇(t) − |u̇(t)|w(t) , − 1 ≤ w(0) ≤ 1, where w is an internal state and ρ > 0, Fc > 0 are constants. A good introductory text on the relationship between the Dahl model and the Coulomb model of dry friction may be found in [8]. The LuGre model is a generalization of the Dahl model introduced in 1995 to include the Stribeck effect, that is the decrease of friction at low velocities [5]. The LuGre model is given by [2]: |u̇(t)| ẋ (t) = −σ0  x (t) + u̇(t), g u̇(t) (1) x(0) = x0 ,  F (t) = σ0 x(t) + σ1 ẋ(t) + f u̇(t) , where t ≥ 0 denotes time; the parameters σ0 > 0 and σ1 > 0 are respectively the stiffness and the microscopic damping friction coefficients; the function g is continuous with g (ϑ) > 0 for all ϑ ∈ R and it represents the macrodamping friction ; x(t) ∈ R is the average deflection of the bristles; x0 ∈ R is the initial state; u(t) is the relative displacement and is the input of the system; F (t) is the friction force and is the output of the system; and f is continuous and such that f (0) = 0. When the function g is constant, σ1 = 0 and f is the zero function, the system (1) reduces to the Dahl model. Both the LuGre and the Dahl models have been used in various applications, see for instance [1, 7, 9]. The main contribution of this paper is Theorem A which is stated in Section 3.1. 3 This theorem provides the analytic description of the minor loop of the LuGre and Dahl models when the input is bimodal like in Figure 2. The paper is organized as follows: Section 2 provides the mathematical notation used in the text. In Section 3 we present and prove the main result which is the analytical description of the minor loop of the LuGre and Dahl models. Section 4 has a pedagogical interest: using numerical simulations we present examples that illustrate the constructive process which leads to the hysteresis and minor loops. Some of these examples are aimed for the reader who may not be familiar with the technicalities that underline the methodology used here. The conclusions are provided in Section 5. 2. Mathematical notation We say that a subset of R is measurable when it is Lebesgue measurable. Consider a function g : I ⊆ R → R where I is an interval. We say that g is measurable if g −1 (B) is a measurable set for any set B in the Borel algebra of R or, equivalently, if {x ∈ I : g(x) > a} is a measurable set for all a ∈ R, [21, 23]. For a measurable function g : I → R, kgk denotes the essential supremum of the function |g| where | · | is the absolute value. We recall that C 0 (R+ , R) denotes the space of continuous functions defined from R+ to R endowed with the norm k · k. Also W 1;∞ (R+ , R) denotes the Sobolev space of absolutely continuous functions u : R+ → R. For this class of functions, the derivative u̇ is measurable, and we have kuk < ∞, ku̇k < ∞. Endowed with the norm kuk1,∞ = max (kuk, ku̇k), W 1;∞ (R+ , R) is a Banach space [15, pp. 280–281]. Finally, L∞ (I, R) denotes the Banach space of measurable functions u : I → R such that kuk < ∞, endowed with the norm k · k. For T > 0 we define ΩT as the set of all T –periodic functions u ∈ W 1;∞ (R+ , R). 3. Main result 3.1. Statement of the main result We consider the LuGre model (1) with an input u ∈ W 1;∞ (R+ , R). In [19] it is proved that for all x0 ∈ R, the differential equation (1) has a unique Carathéodory solution x ∈ W 1;∞ (R+ , R) and that F ∈ L∞ (R+ , R). 4 To present the main result of this work which is the analytic characterization of the minor loop we define formally the set of bimodal inputs needed to generate this minor loop. Definition 1. Let umin,1 , umin,2 , umax,1 , umax,2 ∈ R be such that umin,1 ≤ umin,2 < umax,1 ≤ umax,2 and at least one of the following holds: umin,1 6= umin,2 or umax,1 6= umax,2 . Let t1 , t2 , t3 , t4 ∈ R+ be such that 0 < t1 < t2 < t3 < t4 . We define Mumin,1 ,umin,2 ,umax,1 ,umax,2 ,t1 ,t2 ,t3 ,t4 as the set of all functions u ∈ Ωt4 such that u is strictly increasing on the interval [0, t1 ], strictly decreasing on the interval [t1 , t2 ], strictly increasing on the interval [t2 , t3 ], strictly decreasing on the interval [t3 , t4 ]; and u(0) = umin,1 , u(t1 ) = umax,1 , u(t2 ) = umin,2 , u(t3 ) = umax,2 , u(t4 ) = u(0). Theorem A. Let us consider the LuGre model given by Equations (1) with an input u ∈ Mumin,1 ,umin,2 ,umax,1 ,umax,2 ,t1 ,t2 ,t3 ,t4 . Then the following statements hold: (a) The hysteresis loop that corresponds to the input u is the set   G◦u = ψu (t), y ◦ (t) ∈ R2 , t ∈ [0, %4 ] , where y ◦ is given by σ0  h σ0 i y ◦ (t) = e− g(0) (t−%i ) y ◦ (%i ) − g(0) e g(0) (t−%i ) − 1 , for t ∈ [%i , %i+1 ] and i ∈ {0, 1, 2, 3}, and where σ0 e− g(0) %4  σ0 σ0 σ0 σ0  ◦ % g(0) 1 % g(0) 2 % g(0) 3 % g(0) 4 y (0) = g(0) σ0 2e − 2e + 2e−e − 1 and 1 − e− g(0) %4 h σ0 i σ0 y ◦ (%i ) = e− g(0) (%i −%i−1 ) y ◦ (%i−1 ) + g(0) e g(0) (%i −%i−1 ) − 1 for i ∈ {1, 2, 3, 4}; and ψu is given by    t + umin,1 for t ∈ [0, %1 ],      −t + % + u for t ∈ [%1 , %2 ], 1 max,1 ψu (t) =   t − %2 + umin,2 for t ∈ [%2 , %3 ],      −t + % + u for t ∈ [%3 , %4 ], 3 max,2 being %0 = 0, %1 = umax,1 − umin,1 > 0, %2 = %1 + umax,1 − umin,2 > %1 , %3 = %2 + umax,2 − umin,2 > %2 , and %4 = %3 + umax,2 − umin,1 > %3 . 5 (b) The minor loop that corresponds to the input u is the set   Nu = ψu (t), y ◦ (t) , t ∈ [%1 , %5 ] , where %5 = umax,1 − umin,2 + %2 ∈ (%2 , %3 ]. Comment. Observe that the sets G◦u and Nu are the geometric loci of parametrized curves. Theorem A, thus, gives an explicit parametrization of these curves. 3.2. Proof of Theorem A The proof of Theorem A is done is three steps: Step 1: The hysteresis loop of the LuGre and the Dahl models are derived in Section 3.2.1. Step 2: A normalized input is presented in Section 3.2.2. Step 3: The determination of the equations of the minor loop is done in Section 3.2.3. 3.2.1. Hysteresis loop of the LuGre and the Dahl models To prove Theorem A and therefore to derive the explicit expression of the hys- teresis loop of the LuGre and the Dahl models, we follow the methodology presented in [11, 19]. In this section we recall and adapt the main steps of this methodology. The reader unfamiliar with this theoretical framework is first referred to Example 1 in Section 4.1. Let u ∈ Mumin,1 ,umin,2 ,umax,1 ,umax,2 ,t1 ,t2 ,t3 ,t4 and take T = t4 . Also, take γ ∈ (0, ∞) and consider the linear time-scale change sγ : R → R defined by sγ (t) = t/γ for all t ∈ R. Then u ◦ sγ is γT –periodic. The system (1) for which the input is u ◦ sγ can be written as z ˙ { u ◦ sγ (t) z ˙ { ẋγ (t) = −σ0 z {  xγ (t) + u ◦ sγ (t) , for almost all t ∈ R+ , ˙ g u ◦ sγ (t) xγ (0) = x0 , z ˙ {  Fγ (t) = σ0 xγ (t) + σ1 ẋγ (t) + f u ◦ sγ (t) , for almost all t ∈ R+ . On the other hand,   z ˙ {  t 1 u ◦ sγ (t) = u̇ sγ (t) · ṡγ (t) = u̇ · , γ γ 6 so that we get   1 u̇ t   γ 1γ t ẋγ (t) = −σ0    xγ (t) + u̇ , for almost all t ∈ R+ , 1 g u̇ t γ γ γ γ xγ (0) = x0 ,    1 t Fγ (t) = σ0 xγ (t) + σ1 ẋγ (t) + f u̇ , for almost all t ∈ R+ . γ γ Now, defining ν = t/γ we rewrite these relations in terms of ν obtaining |u̇(ν)| γ ẋγ (γν) = −σ0   xγ (γν) + u̇(ν), for almost all ν ∈ R+ , g γ1 u̇(ν) xγ (0) = x0 , (2)   1 Fγ (γν) = σ0 xγ (γν) + σ1 ẋγ (γν) + f u̇(ν) , for almost all ν ∈ R+ . γ We define the function zγ by the relation zγ = xγ ◦ s1/γ so that  żγ (ν) = ẋγ s1/γ (ν) · ṡ1/γ (ν) = ẋγ (γν) · γ. Substituting in (2) provides: |u̇(t)| żγ (t) = −σ0   zγ (t) + u̇(t), for almost all t ∈ R+ , g u̇(t) γ zγ (0) = x0 , (3)   σ1 u̇(t) yγ (t) = σ0 zγ (t) + żγ (t) + f , for almost all t ∈ R+ , γ γ where yγ = Fγ ◦ s1/γ . For a given γ > 0, the corresponding output-versus-input graph is the set Gu◦sγ =     u ◦ sγ (t), Fγ (t) , t ≥ 0 = u(t), Fγ ◦ s1/γ (t) = yγ (t) , t ≥ 0 . The hysteresis loop of system (3) is the output-versus-input graph obtained for very slow inputs (that is when γ → ∞) in steady state. The next result, which is a straightforward combination of [18, Proposition 5] and [19, Theorem 9], describes the result of this convergence process. Theorem 2 ([18, 19]). The following statements hold: 7 (a) The sequence of functions (yγ )γ>0 converges in the space L∞ (R+ , R) as γ → ∞. Denote yu? = limγ→∞ yγ , then for all t ∈ R+ we have  Z t  σ0 σ0 ? − g(0) ρu (t) ρu (τ ) yu (t) = σ0 e x0 + e g(0) u̇(τ )dτ , (4) 0 Z t ρu (t) = |u̇(τ )|dτ. (5) 0  (b) For any k ∈ N define the function yu,k ? ∈ L∞ [0, T ], R by yu,k ? (t) = yu? (t + kT ), ? for all t ∈ [0, T ]. The sequence of functions (yu,k )k∈N converges in the space  L [0, T ], R as k → ∞. Denote yu = limk→∞ yu,k , then ∞ ◦ ?  ◦ Z t  ◦ σ0 − g(0) ρu (t) yu (0) σ0 ρ u (τ ) yu (t) = σ0 e + e g(0) u̇(τ )dτ for all t ∈ [0, T ]. (6) σ0 0 Moreover, yu◦ (T ) = yu◦ (0). Statement (a) implies that the graphs Gu◦sγ converge in a sense precised in [11,   Lemma 9] to the graph G?u = u(t), yu? (t) , t ≥ 0 as γ → ∞. The hysteresis loop is given by the “steady state” of the parametrized curve G?u which by statement (b) is the set   G◦u = u(t), yu◦ (t) , t ∈ [0, T ] . (7) Moreover, Theorem 2 (b) gives  Z T  σ 0 ρ (T ) − g(0) yu◦ (0) σ0 ρ (τ ) yu◦ (T ) = σ0 e u + e g(0) u u̇(τ )dτ , σ0 0 yu◦ (T ) = yu◦ (0), which leads to σ0 RT σ0 σ0 e− g(0) ρu (T ) e g(0) ρu (τ ) u̇(τ )dτ yu◦ (0) = . 0 σ0 − g(0) ρu (T ) (8) 1−e Equations (6) and (8) provide the analytical expression of the hysteresis loop (7). This expression includes both the major loop and the minor loop. Observe that for the LuGre model neither σ1 nor f intervene in the equation of the hysteresis loop, and only the value g(0) appears in this equation. Also note that Equations (6)–(8) are also valid for the Dahl model since the latter is a particular case of the LuGre model. In Example 2 of Section 4.2 the reader can find a detailed illustration of the concepts presented in this section. 8 3.2.2. The normalized input The hysteresis loop of the LuGre and the Dahl models is given in (7), and it is characterized by the function yu◦ of Theorem 2 (b). Note that we are considering gen- eral input functions u ∈ Mumin,1 ,umin,2 ,umax,1 ,umax,2 ,t1 ,t2 ,t3 ,t4 that may not allow an explicit calculation of the integral present in Equation (6). To get an explicit calculation of that integral we follow the approach of [11] and [19] that leads to the explicit expres- sion of the hysteresis loop by using the so-called normalized input ψu associated to u. The use of the normalized input will give another parametrization of the curve in (7), an explicit one. According to [11], the normalized input associated to u is a piecewise-linear func- tion ψu ∈ W 1;∞ (R+ , R) that satisfies  ψu ρu (t) = u(t) for all t ∈ R+ , (9) Rt where ρu (t) = 0 |u̇(τ )|dτ is the variation function of u. Note that ρu is strictly increasing so that it is invertible, and ρ−1 u is also strictly increasing. From equation (9) it comes that ψu = u ◦ ρ−1 u so that ψu is strictly increasing on the interval [0, %1 ], where %1 = ρu (t1 ). Thus ψ̇u (%) ≥ 0 when % ∈ (0, %1 ) and ψ̇u (%) exists. On the other hand, by [11, Lemma 2], the set on which ψ̇u is not defined or is different from ±1 has measure zero. Thus ψ̇u (%) = 1 for almost all % ∈ (0, %1 ). Using the fact that ψu is absolutely continuous we obtain from the Fundamental Theorem of Calculus that Z % Z % ψu (%) − ψu (0) = ψ̇u (τ ) dτ = dτ = %, for all % ∈ [0, %1 ]. 0 0  Taking into account that ψu ρu (0) = u(0) it comes that ψu (0) = umin,1 so that ψu (%) = % + umin,1 , for all % ∈ [0, %1 ]. The details for the intervals [%1 , %2 ], [%2 , %3 ], and [%3 , %4 ] are given hereafter. • By definition of u we have that u is strictly increasing on the interval [0, t1 ] so that Z t1 Z t1 %1 = ρu (t1 ) = |u̇(t)|dt = u̇(t)dt = u(t1 ) − u(0) = umax,1 − umin,1 . 0 0 Also, from ψu (%) = % + umin,1 for % ∈ [0, %1 ] we get ψu (%1 ) = %1 + umin,1 = umax,1 . 9 Note that ρu is strictly increasing so that it is invertible, and ρ−1 u is also strictly increasing. From equation (9) it comes that ψu = u ◦ ρ−1 u so that ψu is strictly decreasing on the interval [%1 , %2 ], where %2 = ρu (t2 ). Thus ψ̇u (%) ≤ 0 when % ∈ (%1 , %2 ) and ψ̇u (%) exists. On the other hand, by [11, Lemma 2], the set on which ψ̇u is not defined or is different from ±1 has measure zero. Thus ψ̇u (%) = −1 for almost all % ∈ (%1 , %2 ). Using the fact that ψu is absolutely continuous we obtain from the Fundamental Theorem of Calculus that Z % Z % ψu (%) − ψu (%1 ) = ψ̇u (τ ) dτ = −1 dτ = %1 − %, for all % ∈ [%1 , %2 ], %1 %1 which leads to ψu (%) = ψu (%1 ) + %1 − % = umax,1 + %1 − %. • By definition of u we have that u is strictly decreasing on the interval [t1 , t2 ] so that Z t2 Z t1 Z t2 %2 = ρu (t2 ) = |u̇(t)|dt = |u̇(t)|dt + −u̇(t)dt 0 |0 {z } t1 %1 = %1 + u(t1 ) − u(t2 ) = %1 + umax,1 − umin,2 . Also, from ψu (%) = umax,1 + %1 − % for % ∈ [%1 , %2 ] we get ψu (%2 ) = umax,1 + %1 − %2 = umin,2 . From equation (9) it comes that ψu = u ◦ ρ−1 u so that ψu is strictly increasing on the interval [%2 , %3 ], where %3 = ρu (t3 ). Thus ψ̇u (%) ≥ 0 when % ∈ (%2 , %3 ) and ψ̇u (%) exists. On the other hand, by [11, Lemma 2], the set on which ψ̇u is not defined or is different from ±1 has measure zero. Thus ψ̇u (%) = 1 for almost all % ∈ (%2 , %3 ). Using the fact that ψu is absolutely continuous we obtain from the Fundamental Theorem of Calculus that Z % Z % ψu (%) − ψu (%2 ) = ψ̇u (τ ) dτ = dτ = % − %2 , for all % ∈ [%2 , %3 ], %2 %2 which leads to ψu (%) = ψu (%2 ) + % − %2 = umin,2 + % − %2 . 10 • By definition of u we have that u is strictly increasing on the interval [t2 , t3 ] so that Z t3 Z t2 Z t3 %3 = ρu (t3 ) = |u̇(t)|dt = |u̇(t)|dt + u̇(t)dt 0 0 t2 | {z } %2 = %2 + u(t3 ) − u(t2 ) = %2 + umax,2 − umin,2 . Also, from ψu (%) = umin,2 + % − %2 for % ∈ [%2 , %3 ] we get ψu (%3 ) = umin,2 + %3 − %2 = umax,2 . From equation (9) it comes that ψu = u ◦ ρ−1 u so that ψu is strictly decreasing on the interval [%3 , %4 ], where %4 = ρu (t4 ). Thus ψ̇u (%) ≤ 0 when % ∈ (%3 , %4 ) and ψ̇u (%) exists. On the other hand, by [11, Lemma 2], the set on which ψ̇u is not defined or is different from ±1 has measure zero. Thus ψ̇u (%) = −1 for almost all % ∈ (%3 , %4 ). Using the fact that ψu is absolutely continuous we obtain from the Fundamental Theorem of Calculus that Z % Z % ψu (%) − ψu (%3 ) = ψ̇u (τ ) dτ = −1 dτ = %3 − %, for all % ∈ [%3 , %4 ], %3 %3 which leads to ψu (%) = ψu (%3 ) + %3 − % = umax,2 + %3 − %. As a summary, we have    % + umin,1 for % ∈ [0, %1 ],      −% + % + u for ρ ∈ [%1 , %2 ], 1 max,1 ψu (%) = (10)   % − %2 + umin,2 for % ∈ [%2 , %3 ],      −% + % + u for % ∈ [%3 , %4 ], 3 max,2 where %1 = umax,1 − umin,1 > 0, %2 = %1 + umax,1 − umin,2 > %1 , %3 = %2 + umax,2 − umin,2 > %2 , and %4 = %3 + umax,2 − umin,1 > %3 . The function ψu is continuous and %4 –periodic. Its graph in the interval [0, %4 ] is displayed in Figure 3. 11 umax,2 umax,1 umin,2 umin,1 0 ̺6 ̺1 ̺2 ̺5 ̺3 ̺7 ̺4 Figure 3: ψu (%) versus %. 3.2.3. Analytic expression of the hysteresis and minor loops Applying Theorem 2 (b) (Equation (6)) to the particular input ψu , and denoting for simplicity y ◦ := yψ◦ u we obtain that  ◦ Z %  ◦ σ0 − g(0) % y (0) σ0 τ y (%) = σ0 e + e g(0) ψ̇u (τ )dτ for % ∈ [0, %4 ]. σ0 0 Since this expression can be explicitly integrated, we obtain σ0  h σ0 i y ◦ (%) = e− g(0) % y ◦ (0) + g(0) e g(0) % − 1 for % ∈ [0, %1 ], with (11) σ0 e − g(0) %4  σ0 σ0 σ0 σ0  y ◦ (0) = g(0) σ0 − g(0) %4 2e g(0) %1 − 2e g(0) %2 + 2e g(0) %3 − e g(0) %4 − 1 ; 1−e σ0  h σ0 i y ◦ (%) = e− g(0) (%−%1 ) y ◦ (%1 ) − g(0) e g(0) (%−%1 ) − 1 for % ∈ [%1 , %2 ], with (12) σ0  h σ0 i y ◦ (%1 ) = e− g(0) %1 y ◦ (0) + g(0) e g(0) %1 − 1 ; σ 0 (%−% )  h σ0 i ◦ − g(0) 2 ◦ (%−%2 ) y (%) = e y (%2 ) + g(0) e g(0) −1 for % ∈ [%2 , %3 ], with (13) σ0  h σ0 i y ◦ (%2 ) = e− g(0) (%2 −%1 ) y ◦ (%1 ) − g(0) e g(0) (%2 −%1 ) − 1 ; and σ 0 (%−% )  h σ0 i ◦ − g(0) 3 ◦ (%−%3 ) y (%) = e y (%3 ) − g(0) e g(0) −1 for % ∈ [%3 , %4 ], with (14) σ0  h σ0 i y ◦ (%3 ) = e− g(0) (%3 −%2 ) y ◦ (%2 ) + g(0) e g(0) (%3 −%2 ) − 1 . 12 Finally, observe that the hysteresis loop of the LuGre model that corresponds to the input ψu is the set   G◦ψu = ψu (%), y ◦ (%) , % ∈ [0, %4 ] . (15) Taking into account the fact that yu◦ = y ◦ ◦ ρu and u = ψu ◦ ρu it comes from [11,   Lemma 8] that G◦ψu = G◦u = u(t), yu◦ (t) , t ∈ [0, T ] , thus proving statement (a) of Theorem A. To prove statement (b) of Theorem A observe that the minor loop corresponding to the input ψu is the part of the hysteresis loop (15) that corresponds to % ∈ [%1 , %5 ], where Z t5 %5 = ρu (t5 ) = |u̇(t)|dt = %2 + umax,1 − umin,2 ∈ (%2 , %3 ), 0 and where t2 < t5 < t3 is the time such that u(t5 ) = u(t1 ), see Figure 2. This set is the     union of the two arcs ψu (%), y ◦ (%) , % ∈ [%1 , %2 ] and ψu (%), y ◦ (%) , % ∈ [%2 , %5 ] . With this argument, the proof of Theorem A is complete. We remark that the explicit construction of the hysteretic loop, and therefore the identification of the arcs corresponding to the minor loops given in the proof of Theorem A can be generalized to multimodal input functions giving rise to hysteresis loops with many minor loops. This can be done using the normalized input and Equation (6). 4. Examples 4.1. Example 1: an approach to the concept of hysteresis loop A hysteresis system is one for which the output-versus-input graph presents a loop in the steady state for slow inputs [12]. The way to obtain the hysteresis loop that corresponds to a given input is as follows. Consider a periodic input t → u(t). Composing this input with the time-scale change t → t/γ provides a new input uγ : t → u(t/γ). This new input gives rise to an output yγ (t) such that the corre-   sponding output-versus-input graph uγ (t), yγ (t) , t ≥ 0 converges to a fixed curve -the hysteresis loop- in steady state when γ → ∞. Our aim in this section is to illustrate this process using an example. 13 Consider for instance the following system constructed using the Dahl model: ẋγ (t) = u̇γ (t) − |u̇γ (t)|xγ (t), ẏγ (t) = − yγ (t) + xγ (t), (16) xγ (0) = 0, yγ (0) = 0, with input uγ (t) = sin(2πt/γ) and output yγ (t). Figure 4 provides the output-versus-   input graph uγ (t), yγ (t) , t ≥ 0 of system (16) for increasing values of γ. 0.8 0.6 0.4 γ = 2000 0.2 yγ (t) 0 -0.2 γ = 200 -0.4 γ = 20 -0.6 -0.8 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 uγ (t) Figure 4: Output of system (16), yγ versus uγ . Dotted: transient; solid: steady state for γ = 2000; dashed: steady state for γ = 200; dash-dotted: steady state for γ = 20. It can be seen that, as γ → ∞, the steady-state part of the output-versus-input graph converges to a fixed closed curve. This curve is the hysteresis loop of system (16). 14 2 g(ν) 1 0 -10 -5 0 5 10 ν Figure 5: The macrodamping friction function g(ν) versus ν. 4.2. Example 2: the hysteresis loop of the LuGre model The aim of this section is to illustrate the concepts presented in Section 3.2.1 by means of numerical simulations. Following [2], to approximate the Stribeck effect we set: β g(ν) = Fc + (Fs − Fc ) e−|ν/vs | for ν ∈ R, where Fc > 0 is the Coulomb friction force, Fs > 0 is the stiction force, vs > 0 is the Stribeck velocity, and β is a strictly positive constant. The function f is taken to be zero. The values of the different constants are taken to be σ0 = 1, σ1 = 1, Fc = 1, Fs = 2, vs = 1, β = 1, see Figure 5. The input is the continuous 2-periodic piecewise-linear function defined by u(t) = t for t ∈ [0, 1] and u(t) = 2 − t for t ∈ [1, 2]; see Figure 6. Observe that ψu = u. 15 1 u(t) 0.5 0 0 2 4 6 8 t Figure 6: u(t) versus t. We take x (0) = x0 = 0. With these values we obtain yγ by a numerical integration of Equations (3) using Matlab solver ode23s.. Also, using Equations (4)–(5) we obtain yu? . Figure 7 provides the plots yγ (t) versus t for γ = 1, 10, 100 along with the plot yu? (t) versus t. It can be seen that as γ increases, yγ converges to yu? . 1.5 γ=1 1 γ = 10 γ = 100 0.5 y*u(t) yγ (t), yu∗ (t) 0 -0.5 -1 -1.5 0 1 2 3 4 5 6 7 8 t Figure 7: yγ (t) versus t. Dashed γ = 1, dash-dotted γ = 10, dotted γ = 100; solid yu? (t) versus t. ? The functions yu,k ? are given by yu,k (t) = yu? (t + kT ), t ∈ [0, T ], k ∈ N whilst yu◦ is ? calculated from Equations (6) and (8). Figure 8 provides the plots yu,k (t) versus t for ? increasing values of k. It can be seen that yu,k converges to yu◦ as k → ∞. 16 0.8 0.6 k=0 k=1 0.4 k=2 (t), yuo (t) 0.2 0 yu,k ∗ -0.2 k=3 -0.4 -0.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t ? Figure 8: Convergence of the functions yu,k to yu◦ . Dashed yu,k ? (t) versus t for k = 0, 1, 2, 3; solid yu◦ (t) versus t. 0.8 0.6 0.4 yu∗ (t), yuo (t) 0.2 0 -0.2 -0.4 -0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 u(t) Figure 9: Convergence of the output-versus-input graph G?u to the hysteresis loop G◦u . Dotted G?u ; solid G◦u . As in Example 1, the hysteresis loop is the output-versus-input graph obtained for very slow inputs (that is when γ → ∞) in steady state (that is when k → ∞).  For a given γ, the corresponding output-versus-input graph is the set Gu◦sγ = u ◦    sγ (t), Fγ (t) , t ≥ 0 = u(t), Fγ ◦ s1/γ (t) = yγ (t) , t ≥ 0 . Owing to Theorem 2 (a) and to [11, Lemma 9] it comes that the graphs Gu◦sγ converge in a sense detailed   in [11] to the graph G?u = u(t), yu? (t) , t ≥ 0 as γ → ∞. Equations (6) and (8) provide the analytic expression of the hysteresis loop (7). Finally, Figure 9 provides the graph G?u along with the hysteresis loop G◦u . 17 4.3. Example 3: process of convergence that leads to the hysteresis and minor loops The aim of Sections 4.3 and 4.4 is to illustrate Theorem A by means of numerical simulations. Section 4.3 focuses on the process of convergence that leads to the hysteresis loop. Section 4.4 focuses on the variation of the minor loop with the model’s parameters. The function g that characterizes the Stribeck effect is the same as in Section 4.2. Also, the function f is taken to be zero. The input is the continuous ρ4 -periodic piecewise-linear function defined by Equa- tion (10) where umin,1 = 0, umin,2 = 0.2, umax,1 = 1, umax,2 = 1.5, %1 = 1, %2 = 1.8, %3 = 3.1, %4 = 4.6, see Figure 10. Observe that ψu = u and that the normalized variable % is equal to time t. 1.5 1 u(t) 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 t Figure 10: u(t) versus t. We take x (0) = x0 = 0. With these values we obtain yγ by a numerical integration of Equations (3) using Matlab solver ode23s. Also, using Equations (4)–(5) we obtain yu? . Figure 11 provides the plots yγ (t) versus t for γ = 1, 10, 100 along with the plot yu? (t) versus t. It can be seen that as γ increases, yγ converges to yu? . 18 1.5 y*u(t) 1 γ=1 γ = 10 γ = 100 0.5 yu∗ (t), yγ (t) 0 -0.5 -1 -1.5 0 2 4 6 8 10 12 14 t Figure 11: yγ (t) versus t. Dashed γ = 1, dash-dotted γ = 10, dotted γ = 100; solid yu? (t) versus t. ? The functions yu,k ? are given by yu,k (t) = yu? (t + k%4 ), t ∈ [0, %4 ], k ∈ N whilst yu◦ is ? calculated from Equations (6) and (8). Figure 12 provides the plots yu,k (t) versus t ? for increasing values of k. It can be seen that yu,k converges to yu◦ as k → ∞. 1 0.8 k=0 0.6 k=1 0.4 yuo (t), yu,k (t) 0.2 ∗ 0 -0.2 k=2 -0.4 -0.6 -0.8 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 t ? Figure 12: Convergence of the functions yu,k to yu◦ . Dashed yu,k ? (t) versus t for k = 0, 1, 2; solid yu◦ (t) versus t. ?   Figure 13 provides the graphs u(t), yu,k (t) , t ∈ [0, %4 ] for k = 0, 1, 2. It can be   seen that these graphs converge to the hysteresis loop u(t), yu◦ (t) , t ∈ [0, %4 ] . 19 1 0.5 yuo (t), yu,k ∗ (t) 0 hysteresis loop -0.5 k=0 k=1 k=2 -1 0 0.5 1 1.5 u(t)  ?  Figure 13: Convergence of the graphs u(t), yu,k (t) , t ∈ [0, %4 ] to the hysteresis loop   u(t), yu◦ (t) , t ∈ [0, %4 ] . 4.4. Example 4: Variation of the minor loop with the model’s parameters We consider the LuGre model of Section 4.2 with the value σ0 = 6. The input u is the one given in (10) (thus a normalized one) with umin,1 = 0, umin,2 = 0.5, umax,1 = 1, umax,2 = 1.5, with its corresponding values of %i = ti for i = 1, . . . , 5, see Figure 14. The hysteresis loop which is given in Figure 15 is obtained using Equations (11)– (14). Observe that the shape of the minor loop depends greatly on the parameters σ0 , Fs , and on the relative values umin,2 − umin,1 , umax,1 − umin,1 , and umax,2 − umin,1 , see Figures 16 and 17. 1.5 1 u(t) 0.5 0 0 1 2 3 4 t Figure 14: u(t) versus t 20 2 1 y o (t) 0 -1 -2 0 0.5 1 1.5 u(t)  Figure 15: Hysteresis loop G◦u . The marker open circle corresponds to the point u(t1 ), y ◦ (t1 )  whilst the marker rectangle corresponds to the point u(t5 ), y ◦ (t5 ) . The minor loop is plotted in solid line. 0.5 y o (t) 0 -0.5 0 0.5 1 1.5 u(t) Figure 16: Hysteresis loop G◦u for σ0 = 1 (minor loop in solid line). 21 0.5 y o (t) 0 -0.5 0 0.5 1 1.5 u(t) Figure 17: Hysteresis loop G◦u for σ0 = 1 and umin,2 = 0.2 (minor loop in solid line). 5. Conclusions Although the phenomenon of hysteresis has been studied since the second half of the 19th century, the behavior of minor loops as a specific issue did not emerge as a research subject until the second half of the 20th century. The present paper is framed within the increasing interest in the study of the behavior of minor loops. The originality of this work comes from being the first to provide an explicit analytic expression of the minor loops of the LuGre and the Dahl models. Our construction can be generalized to multimodal input functions giving rise to hysteresis loops with many minor loops. The obtained analytic expressions have been illustrated by means of numerical simulations. 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