Classical Prandtl-Ishlinskii Model Research Articles

Nonlinear adaptive tracking control of piezoelectric bimorph actuator using hybrid modeling approach

Piezoelectric bimorph actuator has the advantages of small size, fast response speed and high displacement accuracy, but its inherent hysteresis nonlinearity seriously affect the control accuracy and stability of the system. The dead-zone operator was incorporated into classical Prandtl–Ishlinskii model to enable the description of asymmetric hysteresis of piezoelectric bimorph actuator. A hybrid model approach was developed with neural network and improved Prandtl–Ishlinskii model, and it has the advantages of a neural network with ready-made training algorithms and improve the Prandtl–Ishlinskii (PI) model to describe the asymmetric hysteresis. The adaptive control method was derived from training algorithm of neural network, which can update the weight parameters of Play operator and Dead-zone operator in real time. Comparing the results without control, the RMSE of displacement error decreases by 61.35% with classic model, and decreases by 82.93% with hybrid model and proposed adaptive tracking control. Experimental results show that the proposed hybrid model and adaptive control approach can more effectively compensate the hysteresis of piezoelectric bimorph.

Compensation Method for the Nonlinear Characteristics with Starting Error of a Piezoelectric Actuator in Open-Loop Controls Based on the DSPI Model.

Nanopositioning stages with piezoelectric actuators have been widely used in fields such as precision mechanical engineering, but the nonlinear start-up accuracy problem under open-loop control has still not been solved, and more errors will accumulate, especially under open-loop control. This paper first analyzes the causes of the starting errors from both the physical properties of materials and voltages: the starting errors are affected by the material properties of piezoelectric ceramics, and the magnitude of the voltage determines the magnitude of the starting errors. Then, this paper adopts an image-only model of the data separated by a Prandtl-Ishlinskii model (DSPI) based on the classical Prandtl-Ishlinskii model (CPI), which can improve the positioning accuracy of the nanopositioning platform after separating the data based on the start-up error characteristics. This model can improve the positioning accuracy of the nanopositioning platform while solving the problem of nonlinear start-up errors under open-loop control. Finally, the DSPI inverse model is used for the feedforward compensation control of the platform, and the experimental results show that the DSPI model can solve the nonlinear start-up error problem existing under open-loop control. The DSPI model not only has higher modeling accuracy than the CPI model but also has better performance in terms of compensation results. The DSPI model improves the localization accuracy by 99.427% compared to the CPI model. When compared with another improved model, the localization accuracy is improved by 92.763%.

Temperature-dependent asymmetric Prandtl-Ishlinskii hysteresis model for piezoelectric actuators

A temperature-dependent asymmetric Prandtl-Ishlinskii (TAPI) model is developed to describe changes in hysteresis curves with respect to temperature found in the displacement curves vs. input voltage of a piezoelectric actuator (PEA). The proposed modeling scheme considers nonlinearities in an idealized capacitor term in the electromechanical model of the PEA to introduce both asymmetry and temperature dependence in the model. The developed model has the advantage of incorporating asymmetric and thermal effects in a hysteresis-free region of the model which simplifies inversion of the model as well as parameter determination. A parameter identification scheme is described to simplify model identification, even for a large number of thresholds, based on the advantages of the classical Prandtl-Ishlinskii model. The TAPI model is verified experimentally and a compensator is designed to demonstrate that the PEA output is effectively linearized throughout the temperature range.

Hysteresis modeling and feedforward compensation of a flexible structure actuated by macro fiber composites using bias bipolar Prandtl-Ishlinskii model

As a novel fiber-based piezoelectric composite material, macro fiber composites (MFC) affords notable advantages of good flexibility, high deformability, and large actuation ability. However, the intrinsic hysteresis behavior of the MFC decreases the positioning precision and performance of the flexible structure actuated by MFC actuators. A bias bipolar Prandtl-Ishlinskii (BBPI) model is presented to describe the bias bipolar hysteresis nonlinearity of a MFC-actuated flexible cantilever. The BBPI hysteresis model is composed of two parts: a superposition of the weighted play operators of the classical Prandtl-Ishlinskii (PI) model is employed to characterize the symmetric hysteresis. And a superposition of the weighted dead-zone operators is cascaded to deal with the bias bipolar behavior. Experimental identification results demonstrate that the presented BBPI model exhibits better modeling performance than the classical PI model. A feedforward compensation strategy based on the inverse BBPI hysteresis model is proposed. Experiments on trajectories tracking subject to a triangular wave, sinusoidal wave, and triangular wave with random amplitudes are carried out. Experimental results demonstrate the feasibility and effectiveness of the proposed BBPI model and the inverse feedforward compensator.

Artificial Neural Network Hysteresis Operators for the Identification of Hammerstein Hysteretic Systems

Abstract This paper introduces explicit neural representations of fundamental hysteresis operators such as the play and stop operators. The hysteresis neurons are represented by recurrent artificial neural networks (ANN) using classical activation functions and are trained with gradient-based learning algorithms. The hysteresis neurons are combined into a single ANN hysteretic layer which, from a mathematical point of view, is equivalent to a classical Prandtl-Ishlinskii model expression, but has all the advantages of ANNs. One such benefit is the flexibility it offers to be combined with other ANN layers to obtain various complex model structures. In this paper, this is illustrated by combining it with a linear recurrent neural network to obtain a Hammerstein model structure, which is often considered for the characterization of systems driven by a class of smart material-based actuators. The input nonlinearity in the Hammerstein model is represented by a Prandtl-Ishlinskii ANN hysteretic layer while the linear dynamics are captured using a linear recurrent ANN representing a single-input multiple-output linear time-invariant state-space model. The identification approach of the Hammerstein model with a hysteretic layer is illustrated both numerically and experimentally.

Compensation of Hysteresis in the Piezoelectric Nanopositioning Stage under Reciprocating Linear Voltage Based on a Mark-Segmented PI Model.

The nanopositioning stage with a piezoelectric driver usually compensates for the nonlinear outer-loop hysteresis characteristic of the piezoelectric effect using the Prandtl–Ishlinskii (PI) model under a single-ring linear voltage, but cannot accurately describe the characteristics of the inner-loop hysteresis under the reciprocating linear voltage. In order to improve the accuracy of the nanopositioning, this study designs a nanopositioning stage with a double-parallel guiding mechanism. On the basis of the classical PI model, the study firstly identifies the hysteresis rate tangent slope mark points, then segments and finally proposes a phenomenological model—the mark-segmented Prandtl–Ishlinskii (MSPI) model. The MSPI model, which is fitted together by each segment, can further improve the fitting accuracy of the outer-loop hysteresis nonlinearity, while describing the inner-loop hysteresis nonlinearity perfectly. The experimental results of the inverse model compensation control show that the MSPI model can achieve 99.6% reciprocating linear voltage inner-loop characteristic accuracy. Compared with the classical PI model, the 81.6% accuracy of the hysteresis loop outer loop is improved.

Development of a field dependent Prandtl-Ishlinskii model for magnetorheological elastomers

Magnetorheological elastomers (MREs) offer real-time controllable stiffness and damping properties, and strong hysteresis in the stress-strain responses that depends on magnetic field intensity, strain amplitude and strain rate in a highly nonlinear manner. Prediction of hysteretic stress-strain behavior is essential for effective designs of controllable MRE-based devices. This study presents a stop operator-based Prandtl-Ishlinskii (PI) model for predicting nonlinear hysteresis properties of MREs as functions of the strain amplitude, excitation frequency and magnetic flux density. The stress-strain properties of a MRE fabricated with 40% volume fraction iron particles were experimentally characterized in the shear mode under broad ranges of strain amplitude (2.5–20%), excitation frequency (0.1–50 Hz) and magnetic flux densities (0–450 mT). Subsequently, a stop operator-based classical PI model was formulated considering only 10 hysteresis operators, which required identification of only four parameters. The validity of the classical PI model was assessed using the laboratory-measured data. The proposed classical model is further generalized to enable predictions of MRE dynamic behavior independent of the loading conditions, which would be beneficial for developments in controllable MRE-based adaptive devices. The results demonstrated that the generalized model could accurately characterize nonlinear hysteresis properties of the MRE under the ranges of loading conditions and magnetic field considered.

Compensation of Hysteresis on Piezoelectric Actuators Based on Tripartite PI Model.

Piezoelectric ceramic actuators have been widely used in nanopositioning applications owing to their fast response, high stiffness, and ability to generate large forces. However, the existence of nonlinearities such as hysteresis can greatly deteriorate the accuracy of the manipulation, even causing instability of the whole system. In this article, we have explained the causes of hysteresis based on the micropolarization theory and proposed a piezoelectric ceramic deformation speed law. For this, we analyzed the piezoelectric ceramic actuator deformation speed law based on the domain wall theory. Based on this analysis, a three-stage Prandtl–Ishlinskii (PI) model (hereafter referred to as tripartite PI model) was designed and implemented. According to the piezoelectric ceramic deformation speed law, this model makes separate local PI models in different parts of piezoelectric ceramics’ hysteresis curve. The weighting values and threshold values of the tripartite PI model were obtained through a quadratic programming optimization algorithm. Compared to the classical PI model, the tripartite PI model can describe the asymmetry of hysteresis curves more accurately. A tripartite PI inverse controller, PI inverse controller, and Preisach inverse controller were used to compensate for the piezoelectric ceramic actuator in the experiment. The experimental results show that the inclusion of the PI inverse controller and the Preisach inverse controller improved the tracking performance of the tripartite PI inverse model by more than 80%.

Position control of a single pneumatic artificial muscle with hysteresis compensation based on modified Prandtl-Ishlinskii model.

High-performance position control of pneumatic artificial muscles is limited by their inherent nonlinearity and hysteresis. This study aims to model the length/pressure hysteresis of a single pneumatic artificial muscle and to realize its accurate position tracking control with forward hysteresis compensation. The classical Prandtl-Ishlinskii model is widely used in hysteresis modelling and compensation. But it is only effective for symmetric hysteresis. Therefore, a modified Prandtl-Ishlinskii model is built to characterize the asymmetric length/pressure hysteresis of a single pneumatic artificial muscle, by replacing the classical play operators with two more flexible elementary operators to independently describe the ascending branch and descending branch of hysteresis loops. On the basis, a position tracking controller, which is composed of cascade forward hysteresis compensation and simple proportional pressure controller, is designed for the pneumatic artificial muscle. Experiment results show that the MPI model can reproduce the length/pressure hysteresis of the pneumatic artificial muscle, and the proposed controller for the pneumatic artificial muscle can track the reference position signals with high accuracy. By modelling the length/pressure hysteresis with the modified Prandtl-Ishlinskii model and using its inversion for compensation, precise position control of a single pneumatic artificial muscle is achieved.

Multivariable classical Prandtl–Ishlinskii hysteresis modeling and compensation and sensorless control of a nonlinear 2-dof piezoactuator

A multivariable approach of modeling and feedforward control of rate-independent hysteresis in multi-dof piezoelectric actuators is proposed in this paper. For that, the classical Prandtl–Ishlinskii (CPI) hysteresis model is extended into multivariable. Then, based on the inverse multiplicative structure and on the multivariable CPI model, a compensator is suggested. The proposed compensator does not require an extra-calculation: As soon as the model is identified, the compensator is obtained by structure. Furthermore, inversion of the model is avoided. Additionally to the hysteresis suppression, the multivariable compensator permits to reduce the cross-couplings between the axes which is not possible with standard techniques. The modeling and the inversion-free compensator are afterward applied to a two-degrees-of-freedom (2-dof) piezoactuator. The extensive investigated experimental tests demonstrate that the strong cross-couplings and the strong hysteresis in the two axes can be substantially reduced and linearized, respectively.

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.

Modeling, identification and compensation of hysteresis nonlinearity for a piezoelectric nano-manipulator

Hysteresis nonlinearity widely exists in piezoelectric actuated nano-positioning applications, which degrades their tracking accuracy and limits their precision positioning applications. This paper presents a novel hysteresis modeling and compensation approach to alleviate the adverse effect of the asymmetric and rate-dependent hysteresis nonlinearity for a piezoelectric transducer actuated servo stage. By integrating a generalized input function with the play operator of the classical Prandtl–Ishlinskii model, a novel polynomial-based rate-dependent Prandtl–Ishlinskii (PRPI) model is proposed to capture the hysteresis behavior of the piezoelectric positioning stage, where a polynomial function of input and a time rate function of input are introduced to formulate the generalized input function. Meanwhile, a new adaptive differential evolution optimization algorithm is developed to identify the parameters of the proposed PRPI hysteresis model. Based on the PRPI hysteresis model with the identified parameters, an inverse feedforward controller is constructed to achieve the accurate tracking motion. Furthermore, the hysteresis compensation error of the proposed PRPI model is theoretically analyzed. Finally, comparative experiments are conducted, and the experimental results provided in this paper demonstrate the effectiveness and superiority of the proposed inverse PRPI model compensation approach.

A hysteresis model for a stacked-type piezoelectric actuator

ABSTRACTThe hysteresis nonlinearity in piezoelectric materials brings difficulties in controlling the systems. In order to mitigate the effect of hysteresis, such nonlinearity needs to be characterized and modeled under different load circumstances. For this purpose, the actuator is modeled in terms of a mass-spring-damper system utilizing the stop operator as one of the operators of the Prandtl-Ishlinskii (PI) model. Merging the structural model with the nonlinear hysteresis model, we observe that the results demonstrate better correspondence to the measured output compared to that of the classical PI model for a wide range of working conditions, i.e., different input frequency and amplitude.

Asymmetric Prandtl-Ishlinskii Hysteresis Model for Giant Magnetostrictive Actuator

An asymmetric Prandtl–Ishlinskii (API) hysteresis model for a giant magnetostrictive actuator (GMA) is proposed in this paper. The classical Prandtl–Ishlinskii (PI) model is analyzed and divided into two parts: linear function and operator summation. To enhance model asymmetry, a polynomial function is used in the API model as the center curve of the hysteresis instead of the linear function. The remaining curve of the hysteresis is modeled by a new operator that provides some basic asymmetric hysteresis. In this manner, the proposed API model requires relatively less operators and fewer parameters to describe the asymmetric hysteresis behavior of the GMA. All parameters of the API model are identified by a standard least square method. Simulation results show that the API model is very successful in formulating an asymmetric hysteresis of the GMA. In addition, it provides better identification results compared with the classical PI model.

A generalized Prandtl-Ishlinskii model for characterizing the rate-independent and rate-dependent hysteresis of piezoelectric actuators.

In this paper, a generalized hysteresis model is developed to describe both rate-independent and rate-dependent hysteresis in piezoelectric actuators. Based on the classical Prandtl-Ishlinskii (P-I) model, the developed model adds a quadratic polynomial and makes other small changes. When it is used to describe rate-independent hysteresis, the parameters of the model are constants, which can be identified by self-adaptive particle swarm optimization. The effectiveness of this rate-independent modified P-I model is demonstrated by comparing simulation results of the developed model and the classic Prandtl-Ishlinskii model. Simulation results suggest that the rate-independent modified P-I model can describe hysteresis more precisely. Compared with the classical P-I model, the rate-independent modified P-I model reduces modeling error by more than 50%. When it is used to describe rate-independent hysteresis, a one-side operator is adopted and the parameters are functions with input frequency. The results of the experiments and simulations have shown that the proposed models can accurately describe both rate-independent and rate-dependent hysteresis in piezoelectric actuators.

An Experimental Research on Generalized Prandtl-Ishlinskii Model for Modeling Asymmetric Hysteresis of a Piezoceramic Actuator

A piezoceramic actuator is widely employed in micropositioning and MEMS. However, the piezoceramic actuators are limited due to the natural hysteresis nonlinearity which affect the accuracy of the actuators in applications. In order to revise the hysteresis nonlinearity, lots of hysteresis models have been proposed such as the Preisach model, the classical Prandtl—Ishlinskii model and so on. While some drawbacks still exist with these models, a generalized hysteresis model for asymmetric hysteresis basing on the classical Prandtl—Ishlinskii model is devised. In the modified model, the exponential functions which contain the amplitude and the frequency of the input voltage and its gain factor are introduced into the NLPO (nonlinearity play operator). As a result, the generalized model in this paper applies to modeling asymmetric hysteresis. This model was identified and simulated using the experimental data by other researchers. At last, the validity and the accuracy of the given model were tested through the experiment of the piezoceramic control.

Modeling and Compensation of Asymmetric Hysteresis Nonlinearity for Piezoceramic Actuators With a Modified Prandtl–Ishlinskii Model

This paper presents a modified Prandtl-Ishlinskii (P-I) (MPI) model for the asymmetric hysteresis description and compensation of piezoelectric actuators. Considering the fact that the classical P-I (CPI) model is only efficient for the symmetric hysteresis description, the MPI model is proposed to describe the asymmetric hysteresis nonlinearity of piezoceramic actuators (PCAs). Different from the commonly used approach for the development of asymmetric P-I models by replacing the classical play operator with complex nonlinear operators, the proposed MPI model still utilizes the classical play operator as the elementary operator, while a generalized input function is introduced to replace the linear input function in the CPI model. By this way, the developed MPI model has a relative simple mathematic format with fewer parameters to characterize the asymmetric hysteresis behavior of PCAs. The benefit for the developed MPI model also lies in the fact that an analytic inverse model of the CPI model can be directly applied for the inverse compensation of the asymmetric hysteresis nonlinearity represented by the developed MPI model in real-time applications. To validate the developed MPI model and the inverse hysteresis compensator, simulation, and experimental results on a piezoceramic actuated platform are presented.

Experimental System Identification, Feed-Forward Control, and Hysteresis Compensation of a 2-DOF Mechanism

Most of the micro/nano manipulation mechanisms and systems are commonly based on flexure-based monolithic structures, and are generally driven by piezoelectric actuators. In the presented work, experimental system identification, 1-DOF trajectory tracking with feed-forward control, and hysteresis compensation are investigated. An experimental research facility with laser interferometry-based sensing and measurement technique is established. System identification experiments were performed on a 2-DOF flexure-based mechanism to investigate its dynamics. The system identification procedure, experimental design, data acquisition, analysis and validation of the identified system are presented in details. A linear sine swept signal is applied to the system as an input and the corresponding response of the system is measured with laser interferometry-based sensing and measurement technique. The experimental results are used to evaluate the transfer function and the first natural frequency of the system in the X and Y axes. Experimental validation data is used to verify the accuracy of the identified model. Further, a feed-forward controller is established to track a 1-DOF smooth multiple-frequency trajectory. For hysteresis compensation, inverse PI (Prandtl–Ishlinskii) model is derived from classical PI model. The parameters of the inverse PI model is estimated and validated with the experimental data. Finally, inverse PI model is directly adopted as a feed-forward controller for hysteresis compensation of piezoelectric actuators.

Tracking control of piezoelectric stack actuator using modified Prandtl–Ishlinskii model

This article presents the development of Prandtl–Ishlinskii hysteresis model and tracking control of piezoelectric stack actuator with severe hysteresis. Classical Prandtl–Ishlinskii model is a linearly weighted superposition of many backlash operators with different threshold and weight values, which inherits the symmetric property of the backlash operator at about the center point of the loop formed by the operators. To describe the asymmetric hysteresis of piezoelectric stack actuators, two modified operators were developed, one for ascending branches and another for descending branches. Based on this modified model, a feedforward controller was designed to compensate the hysteresis. Since the modified model describes the inverse of hysteresis, the feedforward controller and the hysteresis of piezoelectric stack actuator canceled each other. To attenuate the creep effect and reduce tracking error, a feedback controller was proposed to work with the feedforward controller. Experimental results show that this control scheme that combines feedforward and feedback controllers greatly improves the tracking of the piezoelectric actuator and the error is less than 0.15 µm.

Modeling Hysteresis with Inertial-Dependent Prandtl-Ishlinskii Model in Wide-Band Frequency-Operated Piezoelectric Actuator

One of the major problems occurring in many technical applications is the presence of the hysteretic behavior in sensors and actuators, which causes a nonlinear relationship between input and output variables in such devices. Since the nonlinear phenomenon of hysteresis degrades the performance of the piezoelectric materials and piezoelectric drive mechanisms, for example, in positioning control framework, it has to be characterized in order to mitigate the effect of the nonlinearity in the devices. This paper is aimed to characterize and model the hysteresis in typical piezoelectric actuators under load-free and preloaded circumstances incorporating the inertial effect of the system. For this purpose, the piezoelectric actuator is modeled as a mass-spring-damper system, which is expressed in terms of a stop operator as one of the essential yet efficient hysteresis operators in the Prandtl-Ishlinskii (PI) model. The reason of utilizing the stop operator in this study is for the sake of control purposes, as the stop operator plays as the inverse of the play operator in the PI model and can be used in a feed-forward controller scheme to suppress the effect of hysteresis in general control framework. The results reveal that this model exhibits better correspondence to the measurement output compared to that of the classical PI model.