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Modeling of Duhem hysteresis with Riemann- Liouville fractional derivative

In the view of memory effect of hysteresis, this work aims to interpret hysteresis nonlinearities in terms of Riemann-Liouville fractional derivative which is a singular operator with memory and hereditary properties. For this purpose, Duhem hysteresis, a model defined by a first order differential equation, is considered and adapted to a fractional order differential equation. Since the fractional order Duhem hysteresis cannot be solved by an analytical scheme, Grünwald-Letnikov approximation is used to obtain numerical solutions. Thus, the effect of fractional order derivative to Duhem hysteresis is demonstrated with graphics obtained by this approximation and plotting using MATLAB. As a result, it is observed that the fractional order model exhibits hysteresis behavior for the orders that are smaller than 1.


Introduction
Hysteresis phenomenon is an important task of science and technology because of its wide range of applications such as ferromagnetic hysteresis in physics, plastic hysteresis in mechanics, soil-moisture hysteresis in hydrology and etc. Hysteresis is a general name of the memory-based nonlinear relation between input and output signals. "Input" and "output" are the terms of system terminology, therefore their sense varies via the application areas. For instance, magnetic field is the input and magnetic induction or magnetization is the output of ferromagnetic hysteresis; force is the input and displacement is the output for mechanic hysteresis; and strain is the input and stress is the output of elastic hysteresis. These examples can be extended to other types of hysteresis.
Increasing importance of hysteresis bring about the necessity of its mathematical modeling and so substantial amount of book, review articles and papers have been published at this area, see [1][2][3][4]. In the mathematical studies, hysteresis nonlinearities are defined by operators between function spaces. Basic hysteresis operators relay, play and stop are defined for simple hysteresis nonlinearities and composition of these operators with some experimental functions, as Preisach or Prandtl-Ishlinskii models, gives hysteresis models for complex systems. Another commonly used model of hysteresis is developed with differential equation known as Duhem model [5]. However, the studies still continue to obtain the most accurate model which is necessary to explain the physical systems exactly. Recently, hysteresis nonlinearities are analyzed in fractional dynamics, see [6][7][8][9][10][11][12][13][14][15][16].
Fractional order modeling of hysteresis has been more often analyzed for piezoelectric materials in sense of Riemann-Liouville fractional derivative. Fractional derivatives have been used to model hysteresis of ferroelectric materials in [17] in which a fractional derivative term of polarization have been added to take into account the dynamical effects cannot be described by integer order derivative. The model leads to good matching between measured and simulation curves on a large frequency bandwidth. Zhu and Zhou [18] presented a linearized hysteresis force for piezoelectric actuated fast tool servomechanism modeled by a fractional order differential equation. Also, Zhu et al. [19] proposed a differential model of hysteresis which models nonlinear component of displacement for piezoelectric actuators. Finally, Ding et al. [20] characterized the hysteresis of piezoelectric actuators in time and frequency domain by a fractional order model and validated the effectiveness of the model by simulations and experiments. Caputo and Carcione [21] introduced to model the memory formalism in the constitutive equation of anelastic media represents hysteresis and fatigue phenomena by using Caputo fractional derivative. More recently, a new fractional order operator with nonsingular kernel known as Caputo Fabrizio derivative has been also used to model hysteresis nonlinearities in ferromagnetic materials in [22]. Moreover, not only nonlocal operators but also local operators with fractional order are considered to model hysteresis behavior. Naser and Ikhouane [23] characterized the Duhem model of hysteresis with a local operator and identified the consistency class of the model with hysteresis behavior via the range of fractional order parameter  . According to this classification, they concluded that the generalized Duhem model with 1   and the generalized semilinear Duhem model with 01   are not in agreement with the hysteresis property. However, they theoretically showed that all other types are compatible with hysteresis behavior.
In this work, we analyze Duhem model of hysteresis nonlinearity with fractional calculus. We choose Duhem model, because it is a general form of elastic hysteresis, in which the fractional dynamics naturally arise, and suitable for ferromagnetic hysteresis [24,25]. To define a fractional order model of hysteresis, we reconstruct the model with fractional differential equation in sense of Riemann-Liouville derivative. The sketch of the paper can be briefed as follows. Section 1 is dedicated to preliminaries for fractional calculus. Integer order model of Duhem hysteresis is reminded in Section 3. Fractional order Duhem model is presented and solved numerically in Section 4 which also includes simulation results for different values of  . Finally, the concluding remarks are given in Section 5.

Preliminaries
Fractional calculus, concerns the generalization of derivative and integral concepts to non-integer order, is as old as classical calculus. However, intense interest by the scientific community to fractional calculus dates back to the last fifty years in which its wide range of application areas have been explored such as viscoelasticity, diffusion phenomena, signal processing, control design and so on. Since different non-integer order derivative and integral operators have been introduced to model different phenomena in science and engineering, fractional calculus is now recognized as an effective mathematical tool.
In this work, we use Riemann-Liouville fractional derivative (RLFD) to model hysteresis phenomena which can be defined for a time dependent function ( ) [26][27][28].
Also, we use the Grünwald-Letnikov (GL) definition of fractional derivative which can be interpreted as the discrete version of RLFD. For a time dependent function ( ) x  the GL derivative is defined as The finite approximation of the GL derivative will be used to discretesize the RLFD in calculation steps (see [28]).

An integer order Duhem model for hysteresis
Duhem model of hysteresis is defined by an ordinary differential equation constructed by depending on the fact that the output changes its character when the input changes its direction. In this paper, we assume that the input represented by x and the output represented by y are real valued functions of time t and also both of them have piecewise continuous derivatives. Then, the Duhem model is given by where  is a positive constant,  and  are material functions. In detail,  corresponds to the mean slopes of hysteresis branches and  corresponds to the mean difference of the output on hysteresis branches. These functions must also satisfy the following conditions for good agreement with the experimental data, [29]: •  is an odd, monotone increasing, piecewise smooth, real valued function with a derivative   that has a finite limit ( )    for large input.
•  is an even, piecewise continuous, real valued function with a limit, for large input, obeying The Duhem hysteresis is a generalized form of mechanics hysteresis backlash which is used for elastic models. Colemann and Hodgdon [24,25] showed that it is also useful to model ferromagnetic hysteresis by choosing the function  and  to match the experimental data.

Fractional order Duhem model for hysteresis
To obtain a fractional order model of hysteresis, we replace the integer order derivative of the input x and the output y in Eq. ( ) 4 with RLFD operator. We choose order of the RLFD as 0 1.

 
Therefore; we get the following fractional order differential equation: Note that the functions  and  have the same properties given for integer order models in Section 3. Since the chain rule is not valid for RLFD, we prefer to solve the fractional Duhem model numerically. For this purpose, GL approximation of RLFD is used. The solution procedure can be summarized as below. Firstly, the time interval  For graphical representation, we consider zero initial conditions and two types of  and  functions, one is used for elastic hysteresis and the other is used for ferromagnetic hysteresis. Although the positive constant  is calculated from experimental data to match the hysteresis nonlinearity exactly, it can be observed that  is often equal to 1 in both the experimental and the theoretical studies [29,30]. Therefore, we select 1  = to focus the effect of fractional order on the hysteresis nonlinearity and we do not aim to explore different values of this parameter to keep confusion away. Also we choose step size 0.01 h = and all figures are plotted for sinusoidal input functions.
In Figure 1,  Similarly, we investigate fractional order model of ferromagnetic hysteresis with respect to  in Figure 2. Therefore, the  and  functions can be chosen as the following, (see [29]):

Conclusions
We investigated the fractional order model of hysteresis nonlinearity in the sense of Riemann-Liouville fractional derivative. Therefore, the Duhem model of hysteresis was defined by a fractional differential equation. Although the integer model analytically could be solved by using the chain rule of the classical derivative, the same solution strategy could not be applied to the fractional model since the fractional derivative does not supply the chain rule. Thus, a numerical approximation which is Grünwald-Letnikov approach was used to solve fractional Duhem hysteresis. Fractional order models were plotted for two kinds of hysteresis nonlinearities which are used to model elastic and magnetic hysteresis via sinusoidal inputs. It was concluded that the model still shows hysteresis behavior for different values of the order. Also, it was observed from the figures that the fractional derivative affected the shape of hysteresis for both elastic hysteresis and ferromagnetic hysteresis. Moreover, the fractional derivative led to smooth hysteresis nonlinearity for elastic hysteresis.