Mathematical Model Of Hysteresis Research Articles

Modeling and Control of a Linear Piezoelectric Actuator

To improve the output displacement of piezoelectric actuators, a linear piezoelectric actuator based on a multistage amplifying mechanism with a small volume, large thrust, high resolution, high precision, and fast response speed is proposed. However, inherent nonlinear characteristics, such as hysteresis and creep, significantly affect the output accuracy of piezoelectric actuators and may cause system instability. Therefore, a complex nonlinear hysteresis mathematical model with a high degree of fit was established. A Play operator was introduced into the backpropagation neural network, and a genetic algorithm (GA) was used to reduce the probability of the fitting of the neural network model falling into a local minimum. Moreover, simulation and experimental test platforms were constructed. The results showed that the maximum displacement of the actuator was 558.3 μm under a driving voltage of 150 V and a driving frequency of 1 Hz. The complex GA-BP neural network model of the piezoelectric actuator not only exhibited high modeling accuracy but also solved the problems of strong randomness and slow convergence. Compared with other control algorithms, the GA-BP fuzzy PID control exhibited higher control precision.

Development of a hysteresis model based on axisymmetric and homotopic properties to predict moisture transfer in building materials

Current hygrothermal behaviour prediction models neglect the hysteresis phenomenon. This leads to a discrepancy between numerical and experimental results, and a miscalculation of buildings’ durability. In this paper, a new mathematical model of hysteresis is proposed and implemented in a hygrothermal model to reduce this discrepancy. The model is based on a symmetry property between sorption curves and uses also a homotopic transformation relative to a parameter [Formula: see text]. The advantage of this model lies in its ease of use and implementation since it could be applied with the knowledge of only one main sorption curve by considering [Formula: see text], in other words, we only use the axisymmetric property here. In the case where the other main sorption curve is known, we use this curve to incorporate the homotopy property in order to calibrate the parameter [Formula: see text].The full version of the proposed model is called Axisymmetric + Homotopic. Furthermore, it was compared not only with the experimental sorption curves of different types of materials but also with a model that is well known in the literature (CARMELIET’s model). This comparison shows that the Axisymmetric + Homotopic model reliably predicts hysteresis loops of various types of materials even with the knowledge of only one of the main sorption curves. However, the full version of Axisymmetric + Homotopic model is more reliable and covers a large range of materials. The proposed model was incorporated into the mass transfer model. The simulation results strongly match the experimental ones.

Tilt Active Vibration Isolation Using Vertical Pendulum and Piezoelectric Transducer with Parallel Controller

Tilt vibrations inevitably have negative effects on some precise engineering even after applying horizontal and vertical vibration isolations. It is difficult to adopt a traditional passive vibration isolation (PVI) scheme to realize tilt vibration isolation. In this paper, we present and develop a tilt active vibration isolation (AVI) device using a vertical pendulum (VP) tiltmeter and a piezoelectric transducer (PZT). The potential resolution of the VP is dependent on the mechanical thermal noise in the frequency bandwidth of about 0.0265 nrad, which need not be considered because it is far below the ground tilt of the laboratory. The tilt sensitivity of the device in an open-loop mode, investigated experimentally using a voltage controller, is found to be (1.63±0.11)×105 V/rad. To compensate for the hysteresis nonlinearity of the PZT, we experimentally established the multi-loop mathematical model of hysteresis, and designed a parallel controller consisting of both a hysteresis inverse model predictor and a digital proportional–integral–differential (PID) adjuster. Finally, the response of the device working in close-loop mode to the tilt vibration was tested experimentally, and the tilt AVI device showed a good vibration isolation performance, which can remarkably reduce the tilt vibration, for example, from 6.0131 μrad to below 0.0103 μrad.

A new form of rate independence for hysteresis systems

In mathematical texts hysteresis is defined as a rate independent phenomenon, and many mathematical models of hysteresis exhibit this rate independence property. However, experiments suggest that this property is only an approximation of the hysteresis behaviour when the excitation is slow enough. Using a generalized form of the hysteretic Duhem model, we show that, although this model is not rate independent, it still satisfies a new form of rate independence. We also explore the relationship between this new form and the usual property of rate independence.

Current-Cycle Iterative Learning Control for High-Precision Position Tracking of Piezoelectric Actuator System via Active Disturbance Rejection Control for Hysteresis Compensation

As a typical smart structure, the piezoelectric actuator (PEA) is an essential constituent component in piezoelectric-driven positioning stages. Nevertheless, the positioning precision is severely degraded by its innate rate-dependent hysteretic nonlinearity. In this article, an innovative control method which combines active disturbance rejection control (ADRC) and current-cycle iterative learning control (CILC) is proposed by constructing PEA as a second-order disturbance-based structure to handle both the hysteretic nonlinearities and dynamic uncertainties of PEA. The proposed method differs from the prevalent model-inverse solution in hysteresis compensation, where the control performance of the latter relies extremely on the accurateness of the hysteretic model while the former does not require a mathematical model of hysteresis since it is considered as a general disturbance and eliminated. Compared with the existing hysteresis compensation via pure ADRC method, the proposed method has improved robustness by incorporating an additional Iterative learning control (ILC) loop to ADRC. Comparative experimentations are executed on a PEA system and results imply that the proposed approach has better control performance than pure proportional-integral control and ADRC.

Neural network modeling and single-neuron proportional–integral–derivative control for hysteresis in piezoelectric actuators

A new polynomial fitting model based on a neural network is presented to characterize the hysteresis in piezoelectric actuators. As hysteresis is multi-valued mapping, and traditional neural networks can only solve one-to-one mapping, a hysteresis mathematical model is proposed to expand the input of the neural network by converting the multi-valued into one-to-one mapping. Experiments were performed under designed excitation with different driven voltage amplitudes to obtain the parameters of the model using the polynomial fitting method. The simulation results were in good accordance with the measured data and demonstrate the precision with which the model can predict the hysteresis. Based on the proposed model, a single-neuron adaptive proportional–integral–derivative controller combined with a feedforward loop is designed to correct the errors induced by the hysteresis in the piezoelectric actuator. The results demonstrate superior tracking performance, which validates the practicability and effectiveness of the presented approach.

Hysteresis Compensation in Force/Torque Sensors Using Time Series Information

The purpose of this study is to compensate for the hysteresis in a six-axis force sensor using signal processing, thereby achieving high-precision force sensing. Although mathematical models of hysteresis exist, many of these are one-axis models and the modeling is difficult if they are expanded to multiple axes. Therefore, this study attempts to resolve this problem through machine learning. Since hysteresis is dependent on the previous history, this study investigates the effect of using time series information in machine learning. Experimental results indicate that the performance is improved by including time series information in the linear regression process generally utilized to calibrate six-axis force sensors.

Design of the magnetic hysteresis mathematical model based on Preisach theory

In the frame of the constantly increasing in electric power systems (EPSs) complexity, the challenge of ensuring adequacy of relay protection (RP) devices operation becomes more and more urgent. The authors propose to use the detailed mathematical models of the combination «instrument transformers (IT)—RP» with modern EPS simulators as the solution. It is very important to adequately simulate IT, in particular, the magnetic core magnetization process, because IT largely determines the shape of the RP-controlled signal and affects on its operation. However, in practice simplified models are currently used due to the lack of an accurate mathematical description of the IT core magnetization characteristics. Such models do not reflect all processes in the magnetic core. The aim of this work is to develop a hysteresis mathematical model, which will have a high accuracy in reproducing the magnetization processes of transformer core. The main research method is mathematical modeling of the ferromagnetic material magnetization processes in the MathCAD software. The article presents main development principles of the mathematical model with magnetic hysteresis memory based on the Preisach theory, which reproduces with high accuracy both the major and minor hysteresis loops.

Modelling Length/Pressure Hysteresis of a Pneumatic Artificial Muscle using a Modified Prandtl-Ishlinskii Model

Pneumatic artificial muscles have been widely used in various fields owing to their inherent compliance and high power-to-weight ratio. However, the natural hysteresis nonlinearity including length/pressure hysteresis and force/pressure hysteresis degrades their performance in precise tracking control, making it necessary to build a mathematical hysteresis model for hysteresis compensation. This paper deals with the modelling of length/pressure hysteresis of pneumatic artificial muscles. The length/pressure hysteresis loops measured by the isotonic test are found to be asymmetric and independent of the external load when the load is small. Considering that the classical Prandtl-Ishlinskii model is only effective for symmetric hysteresis, a modified Prandtl-Ishlinskii model is proposed to describe the length/pressure hysteresis behaviour. The developed model utilizes two asymmetric operators with simple mathematical forms to independently model the ascending branch and descending branch of hysteresis loops. The model parameters are identified using the recursive least square algorithm. Comparisons between simulation results and experimental measurements demonstrate that the proposed model can characterize the asymmetric major hysteresis loop and minor hysteresis loops with high accuracy.

Information geometry, simulation and complexity in Gaussian random fields

Abstract Random fields are useful mathematical objects in the characterization of non-deterministic complex systems. A fundamental issue in the evolution of dynamical systems is how intrinsic properties of such structures change in time. In this paper, we propose to quantify how changes in the spatial dependence structure affect the Riemannian metric tensor that equips the model's parametric space. Defining Fisher curves, we measure the variations in each component of the metric tensor when visiting different entropic states of the system. Simulations show that the geometric deformations induced by the metric tensor in case of a decrease in the inverse temperature are not reversible for an increase of the same amount, provided there is significant variation in the system's entropy: the process of taking a system from a lower entropy state A to a higher entropy state B and then bringing it back to A, induces a natural intrinsic one-way direction of evolution. In this context, Fisher curves resemble mathematical models of hysteresis in which the natural orientation is pointed by an arrow of time.

Magnetic Properties of Two-Phase Composite Magnetic Material and Its Application to Electrical Equipment

A detailed research on the magnetic properties, preparation, and application of two-phase composite magnetic material was conducted in this paper. Firstly, in order to obtain the characteristics of high remanence and low coercivity, a micro field mathematical model of hysteresis was established and the magnetization model of this material was determined on the basis of micro magnetic theory. Secondly, the relationship between remanence and coercivity was analyzed and the preparation technology of the material was proposed from the perspective of the elemental composition, the heat treatment, and the other steps. Finally, after mastering the magnetization characteristic, conversion and control mechanism of the material, a new power transformer with function of DC bias compensation based on the two-phase composite magnetic material was proposed. The simulation and experimental results showed that the transformer could achieve a good compensation for the DC bias problem by using material remanence, which provides intelligent and energy-saving electrical equipment for the electric network safe operation. DOI: http://dx.doi.org/10.5755/j01.ms.21.4.9707

Design and Implementation of a Novel Controllable Reactor Based on Nanocomposite Magnetic Material

This paper proposes a novel controllable reactor with a function of intrinsic magnetic state regulation based on the nanocomposite magnetic material. First, in order to analyze the conversion mechanism and preparation technology of this material, one mathematical model of hysteresis is established, and the corresponding relationship between the coercivity and the remanence of this material is determined. Second, the nanocomposite magnetic material is implanted into the traditional controllable reactor. Then, the structure design and electromagnetic design of the controllable reactor are completed based on the reluctance of the magnetic circuit theory. Finally, effective control parameters and control system for the controllable reactor are designed. Simulation and experimental results show that the inductance value can be regulated continuously and rapidly, and the reactor can be used for reactive power regulation and harmonic suppression which provides intelligent and energy-saving power equipment for intelligent electric network safe operation.

DC Bias Elimination and Integrated Magnetic Technology in Power Transformer

This paper studies the dc bias phenomenon and integrated magnetic technology in a single-phase transformer. A new transformer with the function of dc bias elimination based on the nano-composite magnetic material is designed, and a controllable reactor is integrated into the transformer by integrated magnetic technology. The mathematical model of hysteresis is established, and the correspondence relationship between coercivity and remanence of this material is determined; dc bias magnetic potential in the transformer core can directionally be eliminated using the conversion of coercivity and remanence of the nano-composite magnetic material. The integrated controllable reactor has the notable characteristics of decoupling with other windings, high linearity, less extra covering area, and energy saving. The prototype experimental results verify the correctness of the design.

Seismic response prediction of base-isolated structures with high damping rubber bearings

To characterize the highly nonlinear hysteretic behavior of high damping rubber (HDR) bearings, a previously developed mathematical hysteresis model, rather than a simplified bilinear hysteresis model, is adopted in this study. Unilateral and bilateral seismic simulation tests of a scaled-down multistory structure isolated with HDR bearings subjected to three recorded earthquakes were conducted. Based on the unilateral test results, different sets of model parameters can be identified. The fidelity of the adopted hysteresis model is investigated by comparing the analytical predictions using all sets of identified parameters with results measured from the tests. The significant influence ascribed to different ground motion characteristics such as the effects of near-field and far-field earthquakes on modeling the hysteretic behavior of HDR bearings is observed. Besides, a feasibility study on applying the identified parameters of the adopted hysteresis model to predict the bilateral seismic responses is performed. Even though the prediction accuracy of the seismic responses is satisfactory in two orthogonally horizontal directions, the bilateral hysteretic modeling of HDR bearings requires further intensive studies considering the coupling effect under bilateral excitations.

Nonlinear Modeling of Magnetic Inductor with Hysteresis in Pspice

A nonlinear modeling of magnetic inductor with hysteresis using PSpice is presented. The describing functions are used to model the high frequency hysteresis to make up the defect of Jile-Atherton static hysteresis mathematical model. The subcircuits which include the nonlinear magnetic inductor are used to the circuit of filter by PSpice. The hysteresis-related characteristic of intermodulation distortion and corner frequency can be predicted by the improved inductor subcircuit models have been verified experimentally.

A systematic method for the development of a three‐phase transformer non‐linear model

AbstractIn this work, a novel three‐phase transformer non‐linear model is developed. The proposed model takes into account the magnetic core topology and the windings connections. The non‐linear characteristic curve of the core material is introduced by its magnetization curve or by its hysteresis loop using the mathematical hysteresis model proposed by Tellinen or the macroscopic hysteresis model proposed by Jiles–Atherton. The eddy currents effects are included through non‐linear resistors using Bertotti's work. The proposed model presents several advantages. An incremental linear circuit, having the same topology with the magnetic circuit of the core, is used in order to directly write the differential equations of the magnetic part of the transformer. The matrix Ld that describes the coupling between the windings of the transformer is systematically derived. The electrical equations of the transformer can be easily written for any possible connection of the primary and secondary windings using the unconnected windings equations and transformation matrices. The proposed methods for the calculation of the coupling between the windings, the representation of the eddy currents and the inclusion of the core material characteristic curve can be used to develop a transformer model appropriate for the EMTP/ATP‐type programs. Copyright © 2009 John Wiley & Sons, Ltd.

Application of different saturation curves in a mathematical model of hysteresis

PurposeThe increase of the accuracy of a mathematical model of hysteresis by the choice of the optimum saturation curve for a given material.Design/methodology/approachHysteresis loops of typical soft magnetic materials are approximated with the help of the Taka´cs magnetization model using different saturation curves. The quality of approximations is determined by the deviation of computed magnetic induction amplitudes, iron losses, apparent remanences and coercivities from the measured values.FindingsBy the proper choice of saturation curve, the relative inaccuracy of approximations can be reduced with reference to the original model based on tangent hyperbolic function.Research limitations/implicationsThe accuracy of approximations worsens close to saturation because of the excessive rise of magnetization due to the linear term of the model. This effect should be minimized by the application of complex saturation curves using greater number of parameters.Practical implicationsOwing to the convenient analytical form and increased accuracy, the model equations can be used in simpler practical evaluations of hysteresis effects and for teaching purposes.Originality/valuePresented form of model equations enables approximation of hysteresis loops and the evaluation of main characteristics of magnetic materials on the basis of any saturation curve.

Hysteresis and creep compensation for piezoelectric actuator in open-loop operation

Piezoelectric actuators are frequently used nowadays in a wide variety of positioning devices. However, a major deficiency of piezoelectric actuators is that their open-loop control accuracy is seriously limited by hysteresis and creep effects. This paper describes a method for simultaneous compensation of the hysteresis and creep of piezoelectric actuator based on an inverse control in open-loop operation. The basis of the inverse control is formed by a hysteresis mathematical model and a creep model, which can describe precisely two phenomena. The method of solving inverse models of hysteresis and creep is described detailedly. Finally, a tracking control experiment of piezoelectric actuators for a desired trajectory is performed according to the proposed method and the experimental results demonstrate that the positioning precision is noticeably improved in open-loop operation compared to the conventional open-loop control without any compensation.

On the classical Preisach model for hysteresis in piezoceramic actuators

A classical Preisach model [Mathematical Models of Hysteresis, Springer, New York, 1991] is developed using a piezoceramic actuator in a stacked form. The classical Preisach model is shown to offer excellent modeling accuracy when the actuator is not subject to any load and is subject to an excitation voltage signal at a low frequency. The accuracy of the Preisach model is shown to increasingly deteriorate as the load being applied to the piezoceramic actuator is increased or the range of frequencies contained in the voltage excitation signal gets wider. The classical Preisach model remains, though, a good model for hysteresis in piezoceramic actuators in applications where the load fluctuation is relatively small and the range of frequencies of the voltage excitation is limited.

A phenomenological mathematical model of hysteresis

This paper starts with the description of a purely mathematical model of the saturation curve and the hysteresis loop based on the fundamental similarities between the Langevin function the specified T(x) function and the sigmoid shape. The T(x) function which is composed of tangent hyperbolic and linear functions with its free parameters can describe the regular anhysteretic magnetisation curve. Developed from this function the model describes not only the regular hysteresis loop but also the biased and other minor loops like the ones produced by the interrupted and reversed magnetisation process and the open “loops” created by a piecewise monotonic magnetising field input of diminishing amplitude. The remanent magnetism as the function of the interrupted field co‐ordinates is predicted by the model in this mathematical form for the first time. The model presented here is based on the principle that all processes follow the shape of the T(x) function describing the shape of the major hysteresis loop of the ferromagnetic specimen under investigation. The model is also applicable to hysteretic processes in other fields.