Consistency of the Duhem Model with Hysteresis

The Duhem model, widely used in structural, electrical, and mechanical engineering, gives an analytical description of a smooth hysteretic behavior. In practice, the Duhem model is mostly used within the following black-box approach: given a set of experimental input-output data, how to tune the model so that its output matches the experimental data. It may happen that a Duhem model presents a good match with the experimental real data for a specific input but does not necessarily keep significant physical properties which are inherent to the real data, independent of the exciting input. This paper presents a characterization of different classes of Duhem models in terms of their consistency with the hysteresis behavior.


Introduction
Hysteresis is a nonlinear behavior encountered in a wide variety of processes including biology, optics, electronics, ferroelectricity, magnetism, mechanics, structures, among other areas.The detailed modeling of hysteresis systems using the laws of Physics is an arduous task, and the obtained models are often too complex to be used in applications.For this reason, alternative models of these complex systems have been proposed [1][2][3][4][5].These models do not come, in general, from the detailed analysis of the physical behavior of the systems with hysteresis.Instead, they combine some physical understanding of the system along with some kind of blackbox modeling.
One of the popular models for hysteresis is the Duhem model proposed in [6].The generalized form of the Duhem model consists of an ordinary differential equation of the form ẋ = (, )( u ), where  is the input and  is the state or the output [7].Other special forms of the model have been used, like the form ẋ =  1 (, ) max{ u , 0} +  2 (, ) min{ u , 0} [8] or the semilinear form ẋ = ( + )( u ) [9].Other important special cases of the Duhem model are the LuGre model of friction [10], the Dahl model of friction [11], and the Bouc-Wen model of hysteresis [12,13].The Duhem model has been used to represent friction [7], electromagnetic behavior [14,15], or hysteresis in magnetorheological dampers [16].
In the current literature, the Duhem model is mostly used within the following black-box approach: given a set of experimental input-output data, how to adjust the Duhem model so that the output of the model matches the experimental data?The use of system identification techniques is one practical way to perform this task.Once an identification method has been applied to tune the Duhem model, the resulting model is considered as a "good" approximation of the true hysteresis when the error between the experimental data and the output of the model is small enough.Then, this model is used to study the behavior of the true hysteresis under different excitations.By doing this, it is important to consider the following remark.It may happen that a Duhem model presents a good match with the experimental real data for a specific input but does not necessarily keep significant physical properties which are inherent to the real data, independent of the exciting input.In the current literature, this issue has been considered in [17,18] regarding the passivity/dissipativity of Duhem model.
In this paper, we investigate the conditions under which the Duhem model is consistent with the hysteresis behavior.The concept of consistency is formalized in [19] where a general class of hysteresis operators is considered.The class of operators that are considered in [19] are the causal ones, with the additional condition that a constant input leads to Lemma 1.Let  ∈  1,∞ (R + , R  ) be nonconstant.Then, there exists a unique function   ∈  1,∞ (  , R  ) that satisfies   I   = .
It is shown in [19] that for hysteresis process, the sequence of functions { I  } >0 converges in  ∞ (  , R  ) as  → ∞.This fact shows that consistency is a mathematical property that any model of hysteresis should satisfy.
For this reason, we consider only nonconstant inputs  in this paper.
Observe that Lemma 3 implies that for any  > 0 there exists a unique function  I  ∈  ∞ (  , R  ) such that  I  I  I  =   (when  = 1, we get   I   = ).The latter equality is equivalent to  I  I   =   .According to Definition 4, the system (1)-( 2) is consistent with respect to (,  0 ) if and only if the sequence of functions  I  converges in  ∞ (  , R  ).
Proposition 5.The system (1)-( 2) is consistent with respect to (,  0 ) in the sense of Definition 4 if and only if the sequence of function Proof.To prove the if part, define the causal operator (5), that   is a sequence of continuous functions.Thus, the function  * is continuous as a uniform limit of continuous function.Lemma 3 implies that there exists a unique continuous function  *  ∈  ∞ (  , R  ) such that  *  I   =  * .Let  ∈   .Since   is continuous, there exists some  ≥ 0 such that  =   ().We get from the relation   =  I  I   that for all  > 0: ∞,  = 0, which means that the system (1)-( 2) is consistent with respect to (,  0 ).
Thus, we have Proposition 5 implies that the consistency of the system (1)-( 2) can be investigated by studying the uniform convergence of the sequence of functions   instead of  I  .Thus, we know from Section 2 that the system (1)-( 2) is a hysteresis only if   converges uniformly as  → ∞.
Problem.In this paper, our objective is to derive necessary conditions and sufficient ones for the uniform convergence of the sequence of functions   as  → ∞.

Classification of Function 𝑔
This section introduces a classification for the function  that is used throughout the paper.
The right and left local fractional derivatives of  at  3 ∈ ( 1 ,  2 ) with respect to order  > 0 are defined respectively as follows [22]: where Γ is the gamma function.
The local fractional derivative of a vector-valued function is the vector of local fractional derivatives of its components.Definition 7. The function  ∈  0 (R, R  ) is said to be of class  > 0 if  (0) = 0 and the quantities   +  (0) and   −  (0) exist, are finite, and at least one of them is nonzero.
where  * ∈  0 (R, R  ) is defined as Proof.Immediate using of the change of variables  = /. Proof.
which contradicts the fact that  is of class  1 .
Proof.see Appendix A.

Necessary Conditions
The objective of this section is to derive necessary conditions for the uniform convergence of the sequence of functions   as  → ∞.
The standard way to ensure that the system (1)-( 2) admits a unique solution is to prove that the right-hand side of (1)-( 2) is Lipschitz with respect to .A function ] :  ⊆ R  × R + → R  is Lipschitz with respect to  if there exists a summable function  : for almost all  ≥ 0 and for all  1 ,  2 ∈ R  that satisfy (,  1 ), (,  2 ) ∈  [21].
Lemma 12. Assume that the system (1)-( 2) has a unique global solution for each input  ∈  1,∞ (R + , R) and initial condition  0 ∈ R  .Assume that the function  is of class  > 0. Suppose that there exists a continuous function  : R for each initial state  0 ∈ R  and each input  ∈  1,∞ (R + , R). Assume that the system (1)-( 2) is consistent with respect to (,  0 ); that is, there exists (ii) one has for all  ≥ 0 that where  * is given in (9).
Proof.By (15), the fact that  ∈  1,∞ (R + , R  ), the continuity of the function , and the relation ‖‖ ∞ = ‖ I   ‖ ∞ , for all  > 0, there exists some  > 0 independent of , and where   is given in (3)-( 4).Thus, On the other hand, we conclude from Lemma 3 that Thus, the continuity of  and , the boundedness of u , and Proposition 11, imply that there exists a constant  > 0 independent of  such that   |(  (), ())( u ()/)| ≤ , for almost all  ≥ 0, for all  > 1.Thus, we can apply the Dominated Lebesgue Theorem [23] to get lim On the other hand, since   is continuous as a uniform limit of continuous sequence of functions, we have When  = 1, we obtain from ( 21) and ( 5) that   () =  0 + ∫  0 (  (), ()) * ( u ()), for all  ≥ 0. Thus, the continuity of the functions  and  * along with the boundedness of the functions   , , and u implies that the function q  is bounded.Therefore,   ∈  1,∞ (R + , R  ) and ( 16) is satisfied.
Example 15.Consider the Following LuGre model [10]: 25) where parameter  is the stiffness,  ∈ R is the average deflection of the bristles and is the output of the system,  0 ∈ R is the initial condition,  ∈  1,∞ (R + , R) is the relative displacement, and is the input of the system.The function  : R → R is defined as where   > 0 is the Coulomb friction force,   > 0 is the stiction force, and V  > 0 is the Stribeck velocity.System ( 25)-( 26) has a unique global solution [21, page 5].On the other hand, the sequence of function   is given by (see (5)) () , for almost all  ≥ 0. ( The following facts are proved in Example 29. (i) There exist  1 ,  > 0 such that |  ()| ≤ , for all  >  1 , where   is the output when we use input  I   instead of  (see system ( 3)-( 4)).
(ii) ‖  −   ‖ ∞ → 0 as  → ∞, where the function Thus, all conditions of Lemma 12 are satisfied.Now, we have to find the value of  and the function  * .We have Thus, the function  ∈  0 (R + , R  ) in ( 25) is of class  = 1 (see Definition 7) and the function  * ∈  0 (R + , R  ) in ( 9) is defined as Therefore, by applying Lemma 12, it follows that the system (29) satisfies (16).

Sufficient Conditions
This section presents sufficient conditions for the uniform convergence of the sequence of functions   as  → ∞ (and hence for consistency of the system (1)-( 2) with respect to (,  0 )).The main results of this section are given in Lemmas 20, 23, and 27.
Definition 16 (see [25]).A continuous function  : The following lemma generalizes Theorem 4.18 in [25, page 172].Indeed, in [25], continuous differentiability is needed, while in Lemma 17, we only need absolute continuity.Also, in [25], the inequality on the derivative of the Lyapunov function is needed everywhere, while in Lemma 17 it is needed only almost everywhere.

Lemma 17. Consider a function
where  > 0 is finite or infinite.Assume the following.
(1) The function  is absolutely continuous on each compact interval of [0, ).

Proof. see Appendix B.
Example 18.We want to study the stability of the following system where  0 and state  take values in R, and input  ∈  1,∞ (R + , R). System (33) has an absolutely continuous solution that is defined on an interval of the form [0, ) [21, page 4].
(1) The function  is absolutely continuous on each compact subset of [0, ).
Although the latter corollary follows immediately from Lemma 17, it is useful in many situations [25].25)- (26), where the solid lines are for   .
Example 22. Consider the following semilinear Duhem model: where  is a Hurwitz  ×  matrix (i.e., every eigenvalue of  has a negative real part), vector  and state  take values in R  .The right-hand side of ( 52) is Lipschitz and thus the system has a unique solution [21].Take an input  ∈  1,∞ (R + , R) such that  −1 (0) = − 0 and that | u ()| ≥  for almost all  ∈ R and for some  > 0.

Class
(2) For the function  in system (1)-( 2), there exist  1 ,  2 ∈ R  such that Then, the sequence of functions   of (5) is independent of  and the operator which maps (,  0 ) to  is consistent.
Proof.By condition (2), the right-hand side of ( 5) is independent of .Thus, the solution   of ( 5) is independent of .
Since  I  I   =   , the function  I  is also independent of  (this is the so-called "rate-independent hysteresis") and hence consistency holds.
Example 24.Consider Bouc's hysteresis model [13] as follows: where Clearly, the function  is of class  = 1 and satisfies condition (2) in Lemma 23.This fact implies that the operator which maps (, (0)) to  is consistent and   is independent of .
Remark 26.Observe that if u is nonzero almost everywhere, then (R \ ) = 0 so that by [26] we have (  (R \ )) = 0 as   is absolutely continuous.An example in which u does not need to be nonzero almost everywhere, is when  is constant on some interval, or on a finite number of intervals, or an infinite number of intervals such that this infinite number has measure zero (e.g., countable).
(ii) There exists a function and hence the consistency of the system (65) is guaranteed with respect to input  and initial condition  0 ).
Example 29.The LuGre model is described by [10] as follows: where parameters ,  1 , and  2 > 0 are, respectively, the stiffness, damping, and viscous friction coefficients,  ∈ R is the average deflection of the bristles,  0 ∈ R is the initial state,  ∈  1,∞ (R + , R) is the relative displacement and is the input of the system, and  is the friction force and is the output of the system.The function  : R → R is defined as where   > 0 is the Coulomb friction force,   > 0 is the stiction force, and V  > 0 is the Stribeck velocity.
The LuGre model can be written in the form of the system (65 Clearly, conditions (67) and (68) are satisfied.Thus, Lemma 27 implies that where the functions   : R + → R and   ∈  1,∞ (R + , R) are defined for all  ≥ 0 as Also, there exist some ,  1 > 0 such that for all  >  1 , the solution of ( 92) is global with ‖  ‖ ∞ ≤ .Now, the following analysis is not a part of Lemma 27, but it follows straightforwardly from it.
Let   be the output of the system when we use the input  I   instead of .We obtain from (93) for almost all  ≥ 0 that which leads to where  , : R + → R  is defined as  , () =   (), for all  > 0 and for all  ≥ 0.
which means that the operator which maps input  and initial state  0 to output  is consistent.The conclusion of the analysis is that the hysteresis loop of the LuGre model is {( q  (), ()),  ≥ 0}, where   is given in (97).Observe that this conclusion has been obtained due to the convergence of   in  1,∞ (R + , R) (see Remark 28).

Conclusion
This paper presented a classification of the possible Duhem models in terms of their consistency with the hysteresis behavior.

Figure 2 :Figure 3 :
Figure 2: (a)   () versus () for different values of .(b)   () versus  for different values of .In each plot, the function   is the solid line.