Review
A survey on hysteresis modeling, identification and control

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Highlights

  • The various mathematical models of hysteresis are surveyed.

  • The applications of hysteresis models are investigated in different areas.

  • Different methods of system identification are considered.

Abstract

The various mathematical models for hysteresis such as Preisach, Krasnosel’skii–Pokrovskii (KP), Prandtl–Ishlinskii (PI), Maxwell-Slip, Bouc–Wen and Duhem are surveyed in terms of their applications in modeling, control and identification of dynamical systems. In the first step, the classical formalisms of the models are presented to the reader, and more broadly, the utilization of the classical models is considered for development of more comprehensive models and appropriate controllers for corresponding systems. In addition, the authors attempt to encourage the reader to follow the existing mathematical models of hysteresis to resolve the open problems.

Introduction

When speaking of hysteresis, one refers to the systems that have memory, where the effects of input to the system are experienced with a certain delay in time. This phenomenon is originated from magnetic, ferromagnetic and ferroelectric materials. It is like the elastic property of materials in which a lag occurs between the application and the removal of a force or field and its subsequent effect. The output of the system cannot be predicted without knowledge about the current state of the hysteretic system.

The importance of this study manifests itself in a mathematical modeling of some systems that involve hysteresis such as in smart materials, magnetic fields or micro-sliding friction where hysteresis is dramatically appeared as compared to other (geometric nonlinear) systems.

Amid the smart materials, piezoceramics, one of the most researched materials, are extensively used in widespread applications involving vibration control [1], adaptive structural shape control [2], [3], [4], structural health monitoring [5], [6], structural acoustic systems [7], [8], [9], hybrid transducers and ultrasonic motors [10], [11], nanopositioning stages [12], [13], and more applications for the reason of high precision, high speed position control, high stiffness and fast response. They also are able to endure compressive forces up to several tons, while providing high resolutions and high bandwidth strains simultaneously.

In addition to piezoceramics, shape memory alloys and magnetostrictive actuators are also classified in the category of smart materials. The application of magnetostrictive materials is reported in transducer design [14], acoustic and industrial applications [10], [15], [16], and hybrid transducers [17], [18], [19].

In comparison with piezoceramics and magnetostrictive materials, shape memory alloys (SMA) are relatively new invention of large field of smart materials [20]. SMA is widely used in industrial applications such as vibration attenuation in civil structures [21] and SMA-based microactuator [22], [23], [24], etc.

In spite of large variety of applications assigned for smart materials, they are all subjected to the main source of nonlinearity, namely the hysteresis. This type of nonlinearity might lead to performance degradation specifically in positioning applications. If this phenomenon is neglected, it will give rise to inaccuracy in open loop control and degrades the tracking performance of the actuator. Also, it could cause undesirable oscillations in the system which could even lead to instability in the closed loop.

In contrast to a simple (but incomplete) representation of friction, i.e., the classical Coulomb friction model approximation that defines the friction force only at non-zero relative velocity (v≠0), in fact, micro-sliding displacements are actually observed [25]. When a contacting body is sliding and moving away from a reversal point, the friction force predominantly appears as a function of velocity, similar as presented by the Coulomb model. However, when the motion is reversed, the frictional effects of the mechanism are also determined by displacement function. Therefore, at a certain instance after the motion reversal, the friction behavior depends not only on the velocity, but also on the displacement, where the particular relation between friction and displacement involves a so-called non-local memory hysteresis [25], [26], [27], [28], [29], [30]. This unique behavior attracts many researchers to thoroughly model the behavior of the contact between the two surfaces, which is realized using asperity junctions that can deform elastically or plastically depending on the load and on the displacement and/or the relative velocity of the surfaces [31].

In high precision positioning applications, the effects of friction present in a mechanical system can lead to significant positioning error. In order to compensate the error due to frictional forces, an effective control strategy is a prerequisite. As a consequence of the complex behavior of friction, linear control strategies are generally unsuitable for providing an optimal performance for controlling a motion of systems with friction. If an accurate model of the system is available, a compensation of the error in the system can be made by applying a feedforward command that is equal to and opposite to the instantaneous force.

In this paper, the various types of hysteresis models are investigated as well as their applications in modeling and control. In the remaining part of this study, different types of methods which are utilized for parameter estimation, system identification and system control will be briefly discussed. This paper will be wrapped up by conclusion.

Section snippets

Hysteresis in a nutshell

The presence of hysteresis in ferroelectric-based materials (such as piezoelectric materials) is an important property which creates constitutive nonlinearities in the relation between input fields E (V/m) and stresses σ (N/m2) and output polarization P (C/m2) and strains ε (m/m) as illustrated in Fig. 1. As detailed in [20], hysteresis is directly associated with the non-centro-symmetric structure of ferroelectric compounds and is observed to some degree at essentially all drive levels.

One of

Mathematical models for hysteresis

In order to simulate the hysteresis phenomenon discussed in previous section, some mathematical models have been developed. These models are classified into two types: (1) operator-based (the models which use operators to characterize hysteresis) and (2) differential-based (the models which use differential equation to characterize hysteresis). Our review is commenced by describing four well-known operator-based models, namely (1) Preisach, (2) Krasnosel'skii–Pokrovskii (KP), (3)

System identification strategies in hysteresis characterization

Most of the models considered in the previous section consist of many parameters to build the shape of the hysteresis curve. In the first place, a suitable model has to be assigned to describe a nonlinear behavior of the system properly, and then the parameters of the proposed model have to be estimated. This matter can be considered from two different points of views. In one hand, an identifier can be designed and substituted to the model of the system for imitating the behavior of the real

Control strategies in hysteretic systems

The compensation control for nonlinear hysteresis systems has been reported in literature for wide ranges of application from piezoelectric actuations, micro-sliding friction, magnetorheological and magnetic damper, nanopositioning systems, shape memory alloy wires to medical devices with tendon-sheath mechanisms. In general, we can classify two main approaches for such compensator, namely (i) open-loop control with no feedback from the output and (ii) close-loop control with availability of

Conclusion

In this study, the various types of mathematical models of hysteresis were surveyed. This paper was organized to illustrate two classes of hysteresis models, namely the operator-based model and the differential-based model which are introduced to the reader. In the second part of the paper, the author addressed the specific issue like the several methods utilized for parameter estimation of the proposed models. Implementations of the presented models are subsequently discussed in the last

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