Prandtl-Ishlinskii Hysteresis Operator Research Articles

A fast and increment independent technique for continuous calculation of the strain energy dissipated during cyclic loading applied to magnesium alloy AZ31

The paper introduces a method for calculating the strain energy dissipated during cyclic loading based on Prandtl-Ishlinskii hysteresis operators. The dissipated strain energy is calculated continuously throughout the loading procedure, the main advantage of the method being independence from the incremental step size. The Prandtl-Ishlinskii energy model is applied to a rheological model for magnesium alloy AZ31 that correctly simulates the material’s complex response when subjected to arbitrary uniaxial loading at room temperature. This approach gives a fast and accurate value of the dissipated strain energy which in turn can be used for estimating the fatigue life of magnesium alloys.

Simultaneous Displacement and Force Estimation of Piezoelectric Stack Actuators Using Charge and Voltage Measurements

This article is devoted to simultaneous displacement and force self-sensing using a single piezoelectric stack actuator. Values of displacement and force are estimated from measurements of charge and applied voltage. To calibrate the self-sensing algorithm, a parameter identification procedure is presented that accounts for hysteresis under varying displacement and force conditions using a standard Prandtl–Ishlinskii hysteresis operator. Quasi-static validation experiments are carried out on a custom-built test rig that shows low average estimation errors of $0.5 \%$ and $-0.4\%$ with standard deviations of $2.7\%$ and $3.1\%$ in displacement and force, respectively.

Use of Prandtl-Ishlinskii hysteresis operators for Coulomb friction modeling with presliding

Prandtl-Ishlinskii stop-type hysteresis operators allow for modeling elasto-plasticity in the relative stress-strain coordinates including the saturation level of the residual constant-tension flow. This lies in direct equivalence to the force-displacement characteristics of nonlinear Coulomb friction, whose constant average value at unidirectional motion depends on the motion sign only, after the transient presliding phase at each motion reversal. In this work, we analyze and demonstrate the use of Prandtl-Ishlinskii operators for modeling the Coulomb friction with presliding phase. No viscous i.e. velocity-dependent component is considered at this stage, and the constant damping rate of the Coulomb friction is combined with the rate-independent losses of presliding hysteresis. The general case of Prandtl-Ishlinskii operator with a continuous distribution function is considered together with a finite elements case, which is useful for implementation in multiple applications. Finally, identification of parameters is addressed and discussed along with two experimental examples.

Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions

It is well known that the Prandtl-Ishlinskii hysteresis operator is locally Lipschitz continuous in the space of continuous functions provided its primary response curve is convex or concave. This property can easily be extended to any absolutely continuous primary response curve with derivative of locally bounded variation. Under the same condition, the Prandtl-Ishlinskii operator in the Kurzweil integral setting is locally Lipschitz continuous also in the space of regulated functions. This paper shows that the Prandtl-Ishlinskii operator is still continuous if the primary response curve is only monotone and continuous, and that it may not even be locally Holder continuous for continuously differentiable primary response curves.

English

Certain financial market strategies are known to exhibit a hysteretic structure similar to the memory observed in plasticity, ferromagnetism, or magnetostriction. The main difference is that in financial markets, the spontaneous occurrence of discontinuities in the time evolution has to be taken into account. We show that one particular market model considered here admits a representation in terms of Prandtl-Ishlinskii hysteresis operators, which are extended in order to include possible discontinuities both in time and in memory. The main analytical tool is the Kurzweil integral formalism, and the main result proves the well-posedness of the process in the space of right-continuous regulated functions.

Kurzweil integral representation of interacting Prandtl-Ishlinskii operators

We consider a system of operator equations involving play and Prandtl-Ishlinskii hysteresis operators. This system generalizes the classical mechanical models of elastoplasticity, friction and fatigue by introducing coupling between the operators. We show that under quite general assumptions the coupled system is equivalent to one effective Prandtl-Ishlinskii operator or, more precisely, to a discontinuous extension of the Prandtl-Ishlinskii operator based on the Kurzweil integral of the derivative of the state function. This effective operator is described constructively in terms of the parameters of the coupled system. Our result is based on a substitution formula which we prove for the Kurzweil integral of regulated functions integrated with respect to functions of bounded variation. This formula allows us to prove the composition rule for the generalized (discontinuous) Prandtl-Ishlinskii operators. The composition rule, which underpins the analysis of the coupled model, then establishes that a composition of generalized Prandtl-Ishlinskii operators is also a generalized Prandtl-Ishlinskii operator provided that a monotonicity condition is satisfied.

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.

A Polynomial-Based Linear Mapping Strategy for Feedforward Compensation of Hysteresis in Piezoelectric Actuators

A set of memory-based properties is employed in this paper for modeling multiple-path hysteresis response of piezoelectric actuators. These properties, namely, targeting turning points, curve alignment, and wiping-out effect, are applied in a linear mapping strategy to develop a mathematical framework for modeling the hysteresis phenomenon. More specifically, the locations of turning points are detected and recorded for the prediction of future hysteresis trajectory. An internal trajectory is assumed to follow a multiple-segmented path via a continuous connection of several curves passing through every two consequent turning points. These curves adopt their shapes via a linear mapping strategy from the reference hysteresis curves with polynomial configurations. Experimental implementation of the proposed method demonstrates slight improvement over the widely used Prandtl–Ishlinskii hysteresis operator. However, to maintain the level of precision during the operation, a sufficient number of memory units must be included to record the turning points. Otherwise, in the event of memory saturation, two memory-allocation modes, namely, “open” and “closed” strategies, can be implemented. It is shown that the closed memory-allocation strategy demonstrates better performance by keeping the most important target points. The proposed modeling framework is adopted in an inverse model-based control scheme for feedforward compensation of hysteresis nonlinearity. The controller is experimentally implemented on a three-dimensional nanopositioning stage for surface topography tracking, a problem typically encountered in scanning probe microscopy applications.

Real-time calibration of open-loop piezoelectric actuators for interferometric applications

In the present paper a piezoelectric actuator realized for interferometric applications is described, together with a numerical model to simulate its electro-mechanical behavior. The actuator is an open-loop device made up of three piezoelectric ceramics glued into a stainless steel case and connected directly to the parallel port of a personal computer by control electronics developed on purpose. It consists of a 16 bit digital-to-analog converter whose voltage is fixed by the parallel port, a charge amplifier which provides the voltage to the piezoelectric ceramics and a current divider for the control of the voltage on each channel. The layout based on three active elements has allowed to obtain a device which is able to perform a straight expansion with a negligible tilting, a desirable feature for an actuator used for interferometric applications. The hysteretic behavior, a typical characteristic of this kind of actuator, was simulated by a numerical model, based on the Prandtl–Ishlinskii hysteresis operator, which shows a high capability to predict the input–output response at any level of the input signal, and it is efficient enough for use in real-time applications.

Clamped elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators

We consider a model for one-dimensional transversal oscillations of an elastic-ideally plastic beam. It is based on the von Mises model of plasticity and leads after a dimensional reduction to a fourth-order partial differential equation with a hysteresis operator of Prandtl-Ishlinskii type whose weight function is given explicitly. In this paper, we study the case of clamped beams involving a kinematic hardening in the stress-strain relation. As main result, we prove the existence and uniqueness of a weak solution. The method of proof, based on spatially semidiscrete approximations, strongly relies on energy dissipation properties of one-dimensional hysteresis operators.

Robust Multiple Frequency Trajectory Tracking Control of Piezoelectrically Driven Micro/Nanopositioning Systems

A novel modeling and control methodology is proposed in this paper for real-time compensation of nonlinearities along with precision trajectory control of piezoelectric actuators in various range of frequency operation. By integrating a modified Prandtl-Ishlinskii hysteresis operator with a second-order linear dynamics, a nonlinear dynamic model and an inverse feedforward controller are developed and experimentally validated for a piezoelectrically driven nanopositioning stage. This modeling and control framework, however, lacks the accuracy due to the hysteresis model limitation, parametric uncertainties, and ever present unmodeled dynamics. Utilizing the sliding mode control strategy coupled with a perturbation estimation technique, a robust controller is then proposed for trajectory tracking of the actuator displacement. The controller gains are adjusted based on an intelligent comparison of the dynamic model and the control law. Eventually, the performance of the proposed controller is verified for the nanopositioning stage which is equipped with a high resolution capacitive position sensor. Experimental results demonstrate that the controller is capable of precisely tracking triangular and multiple frequency sinusoidal trajectories, which are common practices in many scanning probe microscopy systems.

Elastic–ideally plastic beams and Prandtl–Ishlinskii hysteresis operators

AbstractThe one‐dimensional equation for transversal vibrations of an elastoplastic beam is derived from a general three‐dimensional system with a single‐yield tensorial von Mises plasticity model. It leads after dimensional reduction to a multiyield scalar Prandtl–Ishlinskii hysteresis model whose weight function is explicitly given. The use of Prandtl–Ishlinskii operators in elastoplasticity is thus not just a questionable phenomenological approach, but in fact quite natural. The resulting partial differential equation with hysteresis is transformed into an equivalent system for which the existence and uniqueness of a strong solution is proved. The proof employs techniques from the mathematical theory of hysteresis operators. Copyright © 2007 John Wiley & Sons, Ltd.

Modeling, Identification and Compensation of Complex Hysteretic Nonlinearities: A Modified Prandtl-Ishlinskii Approach

Undesired complex hysteretic nonlinearities are present to varying degree in virtually all smart material based sensors and actuators provided that they are driven with sufficiently high amplitudes. This necessitates the development of purely phenomenological models which characterize these nonlinearities in a way which is sufficiently accurate, robust, amenable to control design for nonlinearity compensation and efficient enough for use in real-time applications. To fulfill these demanding requirements, the present paper describes a new compensator design method for invertible complex hysteretic nonlinearities which is based on the so-called modified Prandtl-Ishlinskii hysteresis operator. The parameter identification of this model can be formulated as a quadratic optimization problem which produces the best L 2 2 -norm approximation for the measured input-output data of the real hysteretic nonlinearity. Special linear inequality and equality constraints for the parameters guarantee the unique solvability of the identification problem and the invertibility of the identified model. This leads to a robustness of the identification procedure against unknown measurement errors, unknown model errors and unknown model orders. The corresponding compensator can be directly calculated and thus efficiently implemented from the model by analytical transformation laws. Finally, the compensator design method is used to generate an inverse feedforward controller for a magnetostrictive actuator. In comparision to the conventional controlled magnetostrictive actuator the nonlinearity error of the inverse controlled magnetostrictive actuator is lowered from about 50% to about 3%.

Homogenization of heat equation with hysteresis

The contribution deals with heat equation in the form (c u+W[u]) t= div(a·∇u)+f , where the nonlinear functional operator W[ u] is a Prandtl–Ishlinskii hysteresis operator of play type characterized by a distribution function η. The spatially dependent initial boundary value problem is studied. Proof of existence and uniqueness of the solution is omitted since the proof is a slightly modified proof by Brokate–Sprekels. The homogenization problem for this equation is studied. For ε→0, a sequence of problems of the above type with spatially ε-periodic coefficients c ε , η ε , a ε is considered. The coefficients c *, η * and a * in the homogenized problem are identified and convergence of the corresponding solutions u ε to u * is proved.