Nonlinear diffusion and the Preisach model of hysteresis

Two hysteresis models are investigated for modeling static hysteresis phenomenon in nonoriented magnetic steels. First, the static Preisach model is studied, and two types of experimental data, i.e., first-order reversal curves (FORCs) and concentric hysteresis loops, are employed to identify the weight function. Due to negative values in the obtained experimental weight functions, the resulting Preisach models show nonphysical behavior. To eliminate this nonphysical behavior, an optimization procedure is applied to remove these negative values. The nonphysical behavior is resolved in the modified Preisach model; however, the model still has poor accuracy for asymmetric loops. Alternatively, a new congruency-based static hysteresis model is proposed. In this model, third- or higher-order reversal curves are approximated using measured FORCs and second-order reversal curves (SORCs) by means of a congruency regularity. According to this regularity, any third- or higher-order reversal curve is congruent to SORCs with the same reversal point. A 3-D piecewise interpolation function is employed to interpolate the measured major loop, FORCs, and SORCs. The proposed model results are compared with quasi-static measurements. The comparison reveals that the model is capable of approximating the magnetization curves for the measurements containing complex-shaped B waveforms with a relative energy error of less than 4%.

Problem statement: Piezoelectric actuator is a kind of key driving components for micropositioning stages, micropumps, micro valves, micro gripper and so on in the fields of micro/nano technology such as integrated circuit manufacturing, precision instruments, ultra precision fabrication, biomedical manipulation. It has lots of advantages including high stiffness, fast response times, less heat generating, low power consumption and large force output. But the hysteresis nonlinearity seriously affects working performance of actuators. So a lot of models were proposed to describe the hysteresis nonlinearity. A popular model which was widely used is the Preisach model. In order to obtain accurate displacement output corresponding to arbitrary input voltage with the Preisach model, function output approximation is needed. Approach: In this study, firstly the Preisach model was introduced. Then the function modeling of Preisach model based on a Bayesian Regularization Back Propagation Neural (BRBPNN) was presented and a three layers BPNN was designed. Finally, the BRBPNN was trained in Neural Network toolbox of MATLAB6.0. The Preisach function values not at equal diversion points were calculated by the trained network and the actual displacement outputs and theoretical values corresponding to random voltages input were compared. Results: Experimental results indicate that theoretical displacements and measured displacements agree with very well, the maximum displacement error is 0.35μm and the standard deviation is 0.24 μm. Conclusion: The BRBPNN could realize function approximation in Preisach modeling accurately and could meet the precision requirement in the field of modeling and controlling of piezoelectric actuators.

Compensation of hysteresis nonlinearities in smart material based actuators presents a challenging task for their applications. Many approaches have been proposed in the literature, including the inverse multiplicative scheme. The advantage for such a scheme is to avoid direct model inversions. However, the approach is mainly developed for the Bouc-Wen model. Focusing on the Preisach model which is utilized to describe magnetostrictive actuators, in this paper an inverse compensation approach for Preisach model using the inverse multiplicative structure is developed. Since the input signal is implicitly involved in the Preisach model, it imposes a great challenge to construct the inverse function of the model. To obtain an explicit expression of the input signal from its implicit form so that the inverse multiplicative technique can be applied, the Preisach model is decomposed into a non-memory part and memory part. Using this separation, it only requires to solve the inverse of the non-memory part to obtain an explicit expression of the input signal, thus avoiding constructing the inverse for entire complex dual integral formulation of the Preisach model. Experimental results for a magnetostrictive actuator demonstrate the effectiveness of the proposed approach.

Chapter 2 - Generalized Scalar Preisach Models of Hysteresis

In Part II of this two-part study, system simulations and experimental correlations of a Shape Memory Alloy (SMA) based vibration isolation device (briefly described in Part I) has been presented. This device consists of layers of preconstrained SMA tubes undergoing pseudoelastic transformations under transverse dynamical loading. In Part II, detailed description of the prototype vibration isolation device, its experimental setup, and actual experimental test results are presented. An extensive parametric study has been conducted on a nonlinear hysteretic dynamical system, representing this vibration isolation device utilizing a physically based simplified SMA model and a Preisach model (an empirical model based on system identification) developed in Part I. Both the physically based simplified SMA model and the modified Preisach model have been utilized to perform experimental correlations with the results obtained from actual testing of the device. Based on the investigations, it has been shown that variable damping and tunable isolation response are major benefits of SMA pseudoelasticity. Correlation of numerical simulations and experimental results has shown that large amplitude displacements causing phase transformations of SMA components present in such a device are necessary for effective reduction in the transmissibility of such dynamical systems. It has also been shown that SMA-based devices can overcome performance trade-offs inherent in typical softening spring-damper vibration isolation systems. In terms of numerically predicting the experimental results, it has been shown that the Preisach model gave relatively accurate results due to better modeling of the actual SMA tube behavior. However, for a generic parametric study, the physically based simplified SMA model has been found to be more useful as it is motivated from the constitutive response of SMAs and hence, could easily incorporate different changes in system conditions.

Preisach model has enjoyed extensive applications in describing the hysteresis phenomena. However an important open question in the analysis of hysteresis using Preisach models is the determination of the model parameters. This is to determine the parameter of Preisach function and is referred as the identification problem. However, no general mathematical methods appear to be available for the identification of nonlinear hysteresis system. As a result customized identification algorithms must be developed for each specific area of applications. In order to more closely describe physical systems, it becomes increasingly difficult to compute the Preisach function and its associated parameters from experimental results as the complexity of hysteresis models increases. This paper presents a new approach -- a Wavelet identification of Preisach model. A wavelet can generate a special basis for all functions with finite energy and has some important properties such as orthogonality, symmetry, short support and accuracy. In this application the output of system and the Preisach functions are represented by a series of wavelets. The experiment data is used to determine the coefficients of the base functions. Then the Preisach function and the output of the hysteresis system can be directly calculated by using wavelet methods.

Inverse hysteresis model with magnetic flux density and magnetic field intensity as input and output, respectively, is more preferred, compared with the forward one, for resolving the electromagnetic issues of electrical equipment in terms of magnetic vector potential or with voltage sources. And an inverse hysteresis model with high accuracy and high computation speed has a broader range of applications. However, the well-known classical Preisach hysteresis model is formulated with the forward form, and has the double integration of its distribution function, making it inconvenient and impractical in electrical engineering. In this article, an analytical inverse Preisach model is proposed for the first time. First, the analytical expressions of permeability of the initial magnetization curve, descending and ascending branches of the hysteresis loop are derived using one closed form of the Everett integral function. Subsequently, the analytical inverse Preisach model is derived by the difference method, considering the reversible magnetization component that the classical Preisach model cannot simulate with an odd function analytically. The experiment of one grain-oriented silicon steel sample and one non-oriented silicon steel sample, under different magnetic excitation levels, is conducted, and the accuracy and efficiency of the proposed analytical inverse Preisach model are confirmed by comparing its simulated hysteresis loops with the experimental ones and that computed with the widely used inverse Jiles–Atherton (J–A) hysteresis model.

A control structure is presented for manipulators actuated by joints with pairwise antagonistic pneumatic muscles. The used muscles and the resulting behavior of a single manipulator joint featuring antagonistic muscles in a symmetric configuration are characterized. Pneumatic joint actuation results in a hysteretic behavior, it is shown that in this case the hysteresis can be described by a Preisach hysteresis model. The resulting hysteresis model allows the construction of a model-reference following controller, with a model control loop, designed for good tracking performance and a disturbance rejection loop optimized for suppression of disturbances. Experiments confirm the improvement in tracking control as compared to the system solely controlled by a feedback regulator.

The Preisach model is appropriately modified to describe the clockwise hysteresis with the positive saturation state achieved for some positive input value. It is demonstrated that this type of clockwise hysteresis may naturally appear in front propagation problems such as nonlinear diffusion of electromagnetic fields.

Nanorobotics have been successfully applied to investigate the physical and biomedical properties in the nano scale. As the center piece of nanorobotcis Scanning probe microscope (SPM) has been widely used to measure the topography information of sample surface as well as serving as a manipulation and sensing tool to modify and measure the mechanical and physical properties of sample. However, the manipulation accuracy of the SPM based nanorobot has been considered as a serious issue because of the hysteresis, creep and thermal drift of its piezoelectric actuator. Generally, creep is linear, which is easy to model; thermal drift is unpredictable, however, it is slow and has an ultimate balance state, thus easy to reduce. Unlike creep characteristic, hysteresis is nonlinear and exists ubiquitously in smart material actuators and seriously affects behaviors of them. Thus cancelation of it will significantly improve motion precision using actuator such as piezoelectric actuators. In this research, we proposed a novel hysteresis modeling and compensation strategy: Polynomial Prandtl-Ishlinskii (PPI) operator superposition model, and its inverse compensator. The property of PPI model is that, it possesses a similar modeling capacity as Preisach model, more important, it can be represented as compressive as the classical PI model. The PPI inverse compensator is then combined with a PID controller, named PPI Inverse Aided PID (PIA-PID) controller, to reduce the output error caused by hysteresis, creep and some unknown drift of our atomic force microscope (AFM) based nanorobot.

Hysteresis is a non-linear phenomenon exhibited by the mechanical systems. Beyond elastic limit the loading and unloading path of most of the system will differ and that nonlinear path is indicated by hysteresis. The reason for shape of hysteretic cure may due to either changes in material properties beyond the elastic range or due to the changes in structural geometry because of subjected load. This response is a function of both immediate deformation and the previous residual deformation acted on it since it represents the dissipated energy of structure. The hysteretic characteristics or degrading characteristics includes pinching, stiffness degradation, load deterioration, and sliding. A study of four commonly available hysteresis models, which are Bouc Wen Model, Mostaghel Model, Menegotto Pinto Model and Preisach Model were briefly reviewed and discussed in this section and the outcome of this study is the best fitted model for the nonlinear analysis. The scope of the work is to simulate nonlinear response of the building frame subjected to earthquake excitation in a most effective way.

Stroke maximizing and high efficient hysteresis hybrid modeling for a rhombic piezoelectric actuator

This paper introduces a comparative study of two different hysteresis models, limiting loop model and elemental operator model. The former is regarded as a mathematical approach based on the traditional Preisach model. To circumvent the difficulty of determining the distribution function in the Preisach model, the data of the limiting hysteresis loop and a function transform on the Preisach diagram are employed. The latter is a physical method which adopts a vectorial elemental operator with biaxial anisotropy based on the magnetization mechanisms of the ferromagnetic material. These two models are analyzed and compared in terms of the underlying principle, numerical implementation, and calculation accuracy in this paper. To verify the introduced models, the magnetic properties of non-oriented silicon steel Lycore-140 and soft magnetic composite material Somaloy 500 under alternating excitations are measured and compared with simulated results, which show the validity and practicability of the models but with different calculation accuracy and speed.

The rotor eddy current losses of a high-speed permanent magnet synchronous motor reduce the efficiency of the motor and increase the temperature rise of the rotor. In severe cases, the permanent magnet of the rotor can be demagnetized, affecting the safe operation of the motor. In this paper, an analytical model of rotor eddy current losses calculation based on Maxwell equations is introduced. The eddy current losses of rotor structures, e.g., protection sleeve, shield layer and permanent magnets, are analyzed. The calculation results of rotor eddy current losses are compared with two-dimensional (2D) finite element analysis. The comparison verifies the accuracy and versatility of analytical calculation. Finally, the influences of the variables on eddy current losses in the derived model are analyzed. Based on the principle of minimizing eddy current losses, an optimized structure with copper layers covering both inner and outer surfaces of the protective sleeve is proposed. Furthermore, the optimized distribution parameter of copper film thickness is obtained.

This study presents a novel method for fast calculating the eddy current losses (ECLs) caused by pulse-width modulation (PWM) harmonic voltages in the permanent magnets (PMs) of surface-mounted PM synchronous machines (SPMSMs) with distributed windings or concentrated windings. Based on the small-signal models of the SPMSMs, the functional relationships between the high-frequency harmonic voltages (HFHVs) in the rotor reference frame and the corresponding ECLs are investigated with the time-harmonic finite element analysis (THFEA). Instead of using complex analytical PM ECL models, the relationships are directly established with some numerical methods, such as the Gauss quadrature and the piecewise cubic Hermite interpolation method, based on the results from THFEA. With the functional relationships, the total PM ECL caused by all the HFHVs is simply calculated by summing the ECL generated by each harmonic component in the spectra of the PWM voltage in the rotor reference frame. Both two- and three-dimensional (2D and 3D) models of the SPMSMs are calculated. Besides, to further reduce the needed steps of 3D THFEA, an equivalent 2D THFEA method using modified PM conductivities under different-axis HFHV excitations is proposed. The merits of the proposed method are validated by comparing with the traditional time-stepping finite element method.