Hysteresis Operator Research Articles

Elasto-plastic contact problems with heat exchange

This paper deals with a dynamical model for the motion of a one-dimensional visco-elasto-plastic body in contact with an elasto-plastic obstacle and undergoing thermal expansion. The problem for the unknown displacement and temperature is rewritten in accordance with the formalism of hysteresis operators as solution operators of the underlying variational inequalities. The system consists of the momentum balance and energy balance equations. The existence result for this thermodynamically consistent problem is obtained by using some a priori estimates established for the space discretized problem while the uniqueness result follows from the Lipschitz continuity of the nonlinearities.

Hysteresis operators in metric spaces

Motivated by the sweeping processes, we develop an abstract theory of continuous hysteresis operators acting between rectifiable curves with values in metric spaces. In particular we study the continuity properties of such operators and how they can be extended from the space of Lipschitz continuous functions to the space of rectifiable curves. Applications to the sweeping processes and to the vector play operator are shown.

P.D.E.s with hysteresis 30 years later

Continuous and discontinuous hysteresis operators are first reviewed in general. The Duhem model, the generalized play, the (delayed) relay and the Preisach model are outlined, as well as vector extensions of the two latter models. &nbsp Two examples of initial- and boundary-value problems for P.D.E.s with hysteresis are then illustrated. Well-posedness is proved for quasilinear parabolic problems with either continuous or discontinuous hysteresis. Existence of a weak solution is shown for second-order quasilinear hyperbolic problems with discontinuous hysteresis.

Transfer matrix method for multibody systems for piezoelectric stack actuators

In order to achieve a large displacement output from a piezoelectric actuator, we realized the piezoelectric stack actuator (PSA) by mechanically layering/stacking multi-chip piezoelectric wafers in a series and electrically connecting the electrodes in parallel. In this paper, in order to accurately model the hysteresis and the dynamic characteristics of a PSA, the transfer matrix method for multibody systems (MSTMM) was adopted to describe the dynamic characteristics, and the Bouc-Wen hysteresis operator was used to represent the hysteresis. The vibration characteristics of a PSA and a piezo-actuated positioning mechanism (PPM) are derived and analyzed by the MSTMM; then, the dynamic responses of the PSA and the PPM are calculated. The experimental results show that the new method can accurately portray the hysteresis and the dynamic characteristics of a PSA and a PPM. On one hand, if we use this method to model the dynamic response of the PSA and the PPM, the PSA can be considered as a flexible body, as opposed to a mass-spring-damper system, which is in better agreement with the actual condition. On the other hand, the global dynamics equation is not needed for the study of system dynamics, and the dynamics equation has a small-sized matrix and a higher computational speed. Therefore, this method gives a broad range of possibilities for model-based controller design.

Well-posedness for a class of biological diffusion models with hysteresis effect

This paper is concerned with a class of biological models which consist of nonlinear diffusion equations and a hysteresis operator describing the relationship between some variables of the equations. The existence of solutions to the analogous problem was ever considered by Aiki and Minchev (SIAM J Math Anal 36:2020–2032, 2005) under some assumptions including the global Lipschitz continuity of reaction terms. We show the existence of nonnegative solutions to the problem under consideration using the approximation method when the reaction terms are locally Lipschitz continuous. Moreover, we discuss the continuous dependence of solutions on initial data.

Absolute stability analysis of linear systems with Duhem hysteresis operator

In this paper, we investigate the stability of positive and negative feedback interconnections of a linear system and a Duhem hysteresis operator. We provide sufficient conditions on the linear plant and on the Duhem operator which are based on the counterclockwise (CCW) or clockwise (CW) input–output property of the plant and hysteresis operator. We show the application of our main result in the design of a linear controller to stabilize a simple mechanical system driven by a hysteretic actuator, such as, piezo-actuator or smart material-based actuator.

Control of Magnetic Shape Memory Actuators Using Observer-Based Inverse Hysteresis Approach

Magnetic shape memory (MSM) actuators belong to active material technologies with a high energy density and outstanding field-strain relation. Large-scale multivariate field-stress-strain hysteresis effects, however, antagonize their broad use because of the inherent difficulties with control. In this brief, we describe and compare experimentally several MSM control strategies, including the observer-based inverse hysteresis approach proposed in the previous works and combined here with a linear feedback controller by connecting both in parallel. For a prototypic MSM actuator with return spring, it is shown that the actuator plant can be approximated by an appropriate hysteresis operator and a linear transfer function of residual dynamics. The positioning profiles with a bandwidth 0.1-10 Hz and amplitudes between 0.01 and 0.1 mm have been evaluated by the experiments.

Modeling and Inverse Compensation of Nonmonotonic Hysteresis in VO $_2$-Coated Microactuators

Vanadium dioxide ( <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\rm VO_2}$</tex></formula> ) undergoes a thermally induced solid-to-solid phase transition, which can be exploited for actuation purposes. <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\rm VO_2}$</tex></formula> -coated silicon cantilevers demonstrate abrupt curvature changes when their temperature is varied across the phase transition. Unlike the monotonic hysteresis phenomena observed in many other smart materials, the curvature–temperature hysteresis of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">${\rm VO_2}$</tex></formula> actuators is nonmonotonic due to competing mechanisms associated with the material’s phase transition and the different thermal expansion coefficients of the materials that form the bilayered cantilever. Motivated by the underlying physics, a novel model for the nonmonotonic hysteresis that combines a monotonic Preisach hysteresis operator and a quadratic operator is presented. A constrained least-squares scheme is proposed for model identification, and an effective inverse control scheme is presented for hysteresis compensation. For comparison purposes, a Preisach operator with a signed density function and a single-valued polynomial model are considered. Experimental results show that, for a 300- <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\mu \hbox{m}$</tex></formula> -long actuator, the largest modeling errors with the proposed model, the signed Preisach operator, and the polynomial approximation are 46.8, 80.3, and 483 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\rm m}^{-1}$</tex></formula> , respectively, over the actuated curvature range of [ <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\rm -}$</tex></formula> 104, 1846] <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\rm m}^{-1}$</tex></formula> . In addition, both the largest tracking error and root-mean-square error under the proposed inversion scheme are only around 10% of those under the polynomial-based inversion scheme.

Inversion-free stabilization and regulation of systems with hysteresis via integral action

In this paper, we present conditions for the stabilization and regulation of the tracking error for ann-dimensional minimum-phase system preceded by a Prandtl–Ishlinskii hysteresis operator. A general controller structure is considered; however, we assume that an integral action is present. The common Lyapunov function theorem is utilized together with a Linear Matrix Inequality (LMI) condition to show that, under suitable conditions, the tracking error of the system goes to zero exponentially fast when a constant reference is considered. A key feature of this LMI condition is that it does not require the hysteresis effect to be small, meaning that hysteresis inversion is not required. We use this condition together with a periodicity assumption to prove that a servocompensator-based controller can stabilize the system without using hysteresis inversion. Additionally, we draw parallels between our LMI condition and passivity-based results achieved in the literature. We then verify our LMI results in simulation, where we show that the LMI condition can accurately predict the stability margins of a system with hysteresis. Finally, we conduct experiments using a servocompensator-based controller, where we verify the stability of the system and achieve a mean tracking error of 0.5% for a 200 Hz sinusoidal reference.

Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities

In this paper, we analyze and discuss the well-posedness of two new variants of the so-called sweeping process, introduced by Moreau in the early 70s (Moreau in Sem Anal Convexe Montpellier, 1971) with motivation in plasticity theory. The first new variant is concerned with the perturbation of the normal cone to the moving convex subset $$C(t)$$ C ( t ) , supposed to have a bounded variation, by a Lipschitz mapping. Under some assumptions on the data, we show that the perturbed differential measure inclusion has one and only one right continuous solution with bounded variation. The second variant, for which a large analysis is made, concerns a first order sweeping process with velocity in the moving set $$C(t)$$ C ( t ) . This class of problems subsumes as a particular case, the evolution variational inequalities [widely used in applied mathematics and unilateral mechanics (Duvaut and Lions in Inequalities in mechanics and physics. Springer, Berlin, 1976]. Assuming that the moving subset $$C(t)$$ C ( t ) has a continuous variation for every $$t\in [0,T]$$ t ? [ 0 , T ] with $$C(0)$$ C ( 0 ) bounded, we show that the problem has at least a Lipschitz continuous solution. The well-posedness of this class of sweeping process is obtained under the coercivity assumption of the involved operator. We also discuss some applications of the sweeping process to the study of vector hysteresis operators in the elastoplastic model (Kreja?i in Eur J Appl Math 2:281---292, 1991), to the planning procedure in mathematical economy (Henry in J Math Anal Appl 41:179---186, 1973 and Cornet in J. Math. Anal. Appl. 96:130---147, 1983), and to nonregular electrical circuits containing nonsmooth electronic devices like diodes (Acary et al. Nonsmooth modeling and simulation for switched circuits. Lecture notes in electrical engineering. Springer, New York 2011). The theoretical results are supported by some numerical simulations to prove the efficiency of the algorithm used in the existence proof. Our methodology is based only on tools from convex analysis. Like other papers in this collection, we show in this presentation how elegant modern convex analysis was influenced by Moreau's seminal work.

Homoclinic orbits in a two-patch predator-prey model with Preisach hysteresis operator

Systems of operator-differential equations with hysteresis operators can have unstable equilibrium points with an open basin of attraction. Such equilibria can have homoclinic orbits attached to them, and these orbits are robust. In this paper a population dynamics model with hysteretic response of the prey to variations of the predator is introduced. In this model the prey moves between two patches, and the derivative of the Preisach operator is used to describe the hysteretic flow between the patches. A numerical example of a robust homoclinic loop is presented, and a mechanism creating this homoclinic trajectory is discussed.

Uniqueness of solutions to a mathematical model describing moisture transport in concrete materials

When dealing with concrete materials it is always a big issue how to deal with the moisture transport. Here, we consider a mathematical model for moisture transport, which is given as a system consisting of the diffusion equation for moisture and of the ordinary differential equation which describes a hysteresis operator. In [3] we already proved the existence of a time global solution of an initial boundary value problem of this system, however, the uniqueness is obtained only for one dimensional domains. The main purpose of this paper is to establish the uniqueness of a solution of our problem in three dimensional domains under the assumption of the smooth boundary and initial data.

Dynamics of systems with Preisach memory near equilibria

We consider autonomous systems where two scalar differential equations are coupled with the input-output relationship of the Preisach hysteresis operator, which has an infinite-dimensional memory. A prototype system of this type is an LCR electric circuit where the inductive element has a ferromagnetic core with a hysteretic relationship between the magnetic field and the magnetization. Further examples of such systems include lumped hydrological models with two soil layers; they can also appear as a component of the recently proposed models of population dynamics. We study dynamics of such systems near an equilibrium point. In particular, we show and examine a similarity in the behaviour of trajectories between the system with the Preisach memory operator and a planar slow-fast ordinary differential equation. The nonsmooth Preisach operator introduces a singularity into the system. Furthermore, we classify the robust equilibrium points according to their stability properties. Conditions for stability, instability and partial stability are presented. A robust partially stable point simultaneously attracts many trajectories and repels many trajectories (a behaviour which is not generic for smooth ordinary differential equations). We discuss implications of such local dynamics for the excitability properties of the system.

Mathematical modelling of cerebral blood circulation and cerebral autoregulation: towards preventing intracranial hemorrhages in preterm newborns.

Impaired cerebral autoregulation leads to fluctuations in cerebral blood flow, which can be especially dangerous for immature brain of preterm newborns. In this paper, two mathematical models of cerebral autoregulation are discussed. The first one is an enhancement of a vascular model proposed by Piechnik et al. We extend this model by adding a polynomial dependence of the vascular radius on the arterial blood pressure and adjusting the polynomial coefficients to experimental data to gain the autoregulation behavior. Moreover, the inclusion of a Preisach hysteresis operator, simulating a hysteretic dependence of the cerebral blood flow on the arterial pressure, is tested. The second model couples the blood vessel system model by Piechnik et al. with an ordinary differential equation model of cerebral autoregulation by Ursino and Lodi. An optimal control setting is proposed for a simplified variant of this coupled model. The objective of the control is the maintenance of the autoregulatory function for a wider range of the arterial pressure. The control can be interpreted as the effect of a medicament changing the cerebral blood flow by, for example, dilation of blood vessels. Advanced numerical methods developed by the authors are applied for the numerical treatment of the control problem.

Properties of Solutions of a Control System with Hysteresis

We consider a control system described by two nonlinear related equations. The first equation expresses connection between the input and output of the hysteresis operator, whereas the second one is the diffusion equation. The control constraint is expressed by a multivalued mapping of a phase variable with closed nonconvex values in a finitedimensional space. We also consider a system with a convexified control constraint. We study the existence of a solution, topological properties of the set of admissible “trajectory–control” pairs with different control constraints and clarify relations between the sets of solutions. Bibliography: 28 titles.

Adaptive Control for Uncertain Hysteretic Systems

This paper presents an approach to use the L1 adaptive controller for a class of uncertain systems in the presence of unknown Preisach-type hysteresis in input, unknown time-varying parameters, and unknown time-varying disturbances. The hysteresis operator can be transformed into an equivalent linear time-varying (LTV) system with uncertainties, which means that the effect of the hysteresis can be considered as general uncertainties to the system. Without constructing the inverse hysteresis function, the L1 adaptive control is used to handle the uncertainties introduced by the hysteresis, as well as system dynamics. The adaptive controller presented in this paper ensures uniformly bounded transient and tracking performance for uncertain hysteretic systems. The performance bounds can be systematically improved by increasing the adaptation rate. Simulation results with Preisach-type hysteresis are provided to verify the theoretical findings.

A representation result for hysteresis operators with vector valued inputs and its application to models for magnetic materials

In this work, hysteresis operators mapping continuous vector-valued input functions being piecewise monotaffine, i.e. being piecewise the composition of a monotone with an affine function, to vector-valued output functions are considered.It is shown that the operator can be generated by a unique defined function on the convexity triple free strings. A formulation of a congruence property for periodic inputs is presented and reformulated as a condition for the generating string function.

A novel discrete adaptive sliding-mode-like control method for ionic polymer–metal composite manipulators

Ionic polymer–metal composite (IPMC), also called artificial muscle, is an EAP material which can generate a relatively large deformation with a low driving voltage (generally less than 5 V). Like other EAP materials, IPMC possesses strong nonlinear properties, which can be described as a hybrid of back-relaxation (BR) and hysteresis characteristics, which also vary with water content, environmental temperature and even the usage consumption. Nowadays, many control approaches have been developed to tune the IPMC actuators, among which adaptive methods show a particular striking performance. To deal with IPMCs’ nonlinear problem, this paper represents a robust discrete adaptive inverse (AI) control approach, which employs an on-line identification technique based on the BR operator and Prandtl–Ishlinskii (PI) hysteresis operator hybrid model estimation method. Here the newly formed control approach is called discrete adaptive sliding-mode-like control (DASMLC) due to the similarity of its design method to that of a sliding mode controller. The weighted least mean squares (WLMS) identification method was employed to estimate the hybrid IPMC model because of its advantage of insensitivity to environmental noise. Experiments with the DASMLC approach and a conventional PID controller were carried out to compare and demonstrate the proposed controller’s better performance.

A hybrid model for the play hysteresis operator

A hybrid model is proposed for characterization of the hysteresis nonlinearity of the well-known play operator. The proposed model holds the hysteresis nonlinearity and the memory effects of the play operator. Simulation results are also presented to show the capability of the hybrid model to present the hysteresis nonlinearity with memory effects.

Modeling of Ultra-Precision Piezodriven Positioning System

A neural network based approach for the identification of the rate-dependent hysteresis in the piezoelectric actuators is proposed in this article. In this method, a dynamic hysteresis operator for expanded input space is proposed to extract the change-tendency and rate-dependency of the dynamic hysteresis, the parameters of the hysteretic operator is identified using genetic algorithm. An expanded input space involving the original input variable and the new operator is constructed. Thus, based on the expanded input space, the neural networks can be utilized to approximate the behavior of the rate-dependent hysteresis. Furthermore, the dynamic performance of the model is improved because of the existence of dynamic operator. Finally, the method is used to the modeling of hysteresis in a piezoelectric actuator. The experimental results are presented to verify the effectiveness of the proposed approach.