Hysteresis In Piezoelectric Actuators Research Articles

Modelling and compensation of rate‐dependent hysteresis in piezoelectric actuators based on a modified Madelung model

AbstractBased on the principle of weight superposition, a modified Madelung model is proposed. By combining the signal delay response characteristics (SDRC), a dynamic model and its inverse model are established to describe and compensate for the rate‐dependent hysteresis phenomenon of the piezoelectric actuators (PEAs). The effectiveness of the proposed method is verified by experiments, and the results show that the hysteresis nonlinearity of the PEA is basically eliminated, with the normalized root‐mean‐square error of 0.64% under the excitation of multi‐frequency superposition signal.

Modeling and parameter identification of rate-dependent hysteresis behavior based on modified-generalized Prandtl–Ishlinskii model

The intrinsic characteristic of piezoelectric actuators (PEA), known as hysteresis, has been demonstrated to diminish the capability and stability of the system significantly. This paper proposes a modified-generalized Prandtl–Ishlinskii (MGPI) model to describe the rate-dependent hysteresis in piezoelectric actuators. The developed model incorporates a voltage change rate function to replace the first part of the generalized Prandtl–Ishlinskii (GPI) model. Additionally, the model integrates the cubic polynomial into the envelope function, along with the dynamic thresholds and weights. When describing the hysteresis of the piezoelectric actuator (PEA), the model parameters are identified using the Improved Grey Wolf Optimizer (IGWO) algorithm. To prevent the algorithm from getting trapped in local optima, the cubic chaotic mapping is utilized for population initialization, as well as a nonlinear convergence factor, and the Levy flight strategy factor is introduced to update the Wolf pack’s position. The rate-dependent hysteresis behavior of a PEA under excitation in the 1–200 Hz frequency range was experimentally measured. The measured data were used to demonstrate the validity of the proposed MGPI model. The relative root-mean-square error and the relative maximum error of the MGPI model are 1.41% and 6.00%, respectively, which are lower than those of the GPI model, which are 3.15% and 10.58%. Under the composite frequency driving, the outputs of the GPI model and MGPI model were compared with the measured data of the PEA, the results suggest that the MGPI model and the IGWO algorithm can more accurately describe the rate-dependent hysteresis of the piezoelectric actuators.

Hysteresis modeling and analysis of piezoelectric actuators using a modified LuGre model at different preloads and frequencies

Despite a large of models proposed to characterize the nonlinear hysteresis of piezoelectric actuators, few of them can show the effects of different preloads on the nonlinear hysteresis. In this paper, a modified LuGre model (MLM) is presented for piezoelectric actuators under different preloads to accurately characterize its dynamic hysteresis nonlinearity at different frequencies. Based on the classical LuGre model (CLM), the presented model introduces infinite impulse response filters and smoothness coefficient to simplify it. The parameters of the models were identified by the non-linear least squares method. A series of experiments were conducted to verify the validity of the MLM under different preloads at different frequencies excitation signals.In the constant frequency experiments under different preloads, the MLM reduces the mean error by 68.1%−81.9% compared with the CLM; In the constant preload experiments at different frequencies, the MLM reduces the mean error by 72.2%−83.0% and 65.9%−83.2% compared with the classical Bouc–Wen model (CBM) and the CLM.

Adaptive Backstepping Time Delay Control for Precision Positioning Stage with Unknown Hysteresis

Piezoelectric-actuated precision positioning stages are widely used in high-precision instruments and high-end equipment due to their advantages of high resolution, fast response, and compact size. However, due to the strong nonlinearity of hysteresis, the presence of hysteresis in piezoelectric actuators seriously affects the positioning accuracy of the system. In addition, it is challenging to identify the model parameters for hysteresis. In this paper, an adaptive backstepping time delay control method is proposed for piezoelectric devices system with unknown hysteresis. Firstly, the Bouc–Wen model is used to describe the hysteresis characteristics, and the model is interpreted as a linear term and a bounded uncertain hysteresis term. Then, the time delay estimation technique is used to estimate the hysteresis term of the Bouc–Wen model online, and the unknown parameters of the system and hysteresis model are obtained through adaptive updating laws. Furthermore, the stability of the control scheme is proved based on Lyapunov stability theory. Finally, the effectiveness and superiority of the proposed control scheme are demonstrated by comparing it with two typical hysteresis compensation control algorithms through three different sets of input signals.

Parameter identification of Duhem model based on antlion-fish swarm hybrid algorithm and robust backstepping sliding mode control for rate-dependent hysteresis of piezoelectric actuators

A new parameter identification method of Duhem model based on antlion-fish swarm hybrid algorithm is proposed to characterize the rate-dependent hysteresis. The unknown piecewise functions in Duhem model is identified by a hybrid algorithm. Antlion optimization (ALO) algorithm and artificial fish (AF) swarm algorithm are switched in order of execution and their iterative data are fused to form the antlion-fish swarm hybrid algorithm. The comparison experiment shows the hybrid algorithm has the advantages of fast convergence and high precision. In order to compensate the rate-dependent hysteresis of PEA, a robust backstepping sliding mode feedback controller with feedforward inverse Duhem model compensator are designed. The comparative experimental results of displacement tracking show that this method has good tracking performance. The practicability and effectiveness of the method are verified.

Application of Least-Squares Support-Vector Machine Based on Hysteresis Operators and Particle Swarm Optimization for Modeling and Control of Hysteresis in Piezoelectric Actuators

Nanopositioning systems driven by piezoelectric actuators are widely used in different fields. However, the hysteresis phenomenon is a major factor in reducing the positioning accuracy of piezoelectric actuators. This effect makes the task of accurate modeling and position control of piezoelectric actuators challenging. In this paper, the learning and generalization capabilities of the model are efficiently enhanced to describe and compensate for the rate-independent and rate-dependent hysteresis using a kernel-based learning method. The proposed model is inspired by the classical Preisach hysteresis model, in which a set of hysteresis operators is used to address the problem of mapping, and then least-squares support-vector machines (LSSVM) combined with a particle swarm optimization (PSO) algorithm are used for identification. Two control schemes are proposed for hysteresis compensation, and their performance is evaluated through real-time experiments on a nanopositioning platform. First, an inverse model-based feedforward controller is designed based on the LSSVM model, and then a combined feedback/feedforward control scheme is designed using a classical control strategy (PID) to further enhance the tracking performance. For performance evaluation, different datasets with a variety of hysteresis loops are used during the simulation and experimental procedures. The results show that the proposed method is successful in enhancing the generalization capabilities of LSSVM training and achieving the best tracking performance based on the combination of feedforward control and PID feedback control. The proposed control scheme outperformed the inverse Preisach model-based control scheme in terms of both positioning accuracy and execution time. The control scheme that uses the LSSVM based on nonlinear autoregressive exogenous (NARX) models has significantly less computational complexity compared to our control scheme but at the expense of accuracy.

Robust tracking for nanopositioning stages using sliding mode control with active disturbance rejection: Design and implementation

This paper presents the design and implementation of a novel sliding mode control integrated with active disturbance rejection (SMCDR) for precise robust trajectory tracking of piezoelectric nanopositioning stages. The model uncertainties, nonlinearity, and external disturbances of the piezoelectric nanopositioning stage are regarded as a lumped disturbance, which is estimated by an extended state observer. The active disturbance rejection control (ADRC) strategy is used to realize a preliminary trajectory tracking, while the sliding mode control is adopted to handle the estimation error and residual uncertainties, and to improve the tracking performance. The exponential stability of the proposed SMCDR is proved using Lyapunov’s direct method. The proposed SMCDR controller combines the strength of both ADRC and sliding mode control, exhibits a simple structure, requires only the position measurements, and does not require any information of model parameters except for an approximate constant gain. Experimental results reveal the robustness of the SMCDR approach in suppressing the hysteresis of piezoelectric actuator, and illustrate the superior performance over conventional ADRC, integral sliding mode controller (ISMC) and PID controller for trajectory tracking control of piezoelectric nanopositioning stages.

Prescribed Adaptive Backstepping Control of Nonlinear Systems Preceded by Hysteresis in Piezoelectric Actuators

Prescribed Adaptive Backstepping Control of Nonlinear Systems Preceded by Hysteresis in Piezoelectric Actuators

Identification of Preisach Model Parameters Based on an Improved Particle Swarm Optimization Method for Piezoelectric Actuators in Micro-Manufacturing Stages.

The Preisach model is a typical scalar mathematical model used to describe the hysteresis phenomena, and it attracts considerable attention. However, parameter identification for the Preisach model remains a challenging issue. In this paper, an improved particle swarm optimization (IPSO) method is proposed to identify Preisach model parameters. Firstly, the Preisach model is established by introducing a Gaussian−Gaussian distribution function to replace density function. Secondly, the IPSO algorithm is adopted to Fimplement the parameter identification. Finally, the model parameter identification results are compared with the hysteresis loop of the piezoelectric actuator. Compared with the traditional Particle Swarm Optimization (PSO) algorithm, the IPSO algorithm demonstrates faster convergence, less calculation time and higher calculation accuracy. This proposed method provides an efficient approach to model and identify the Preisach hysteresis of piezoelectric actuators.

A modified direct inverse Prandtl–Ishlinskii model based on two sets of operators for the piezoelectric actuator hysteresis compensation

Piezoelectric actuator (PEA) is widely applied in the field of micro/nano high precision positioning. However PEA has the phenomenon of hysteresis non-linearity between input voltage and output displacement, due to the natural property of piezoelectric materials. The PEA hysteresis can be compensated by hysteresis models, which makes the input voltage and output displacement more linearity. The research work on compensation of PEA hysteresis by using various hysteresis models has been being a hot topic. This paper presents a modified direct inverse rate-independent Prandtl–Ishlinskii (PI) (MDIPI) model for compensating the hysteresis of PEA. The proposed MDIPI model has two different sets of operators compared with classical PI (CPI) model having one set of operators. For the two sets operators in MDIPI model one is rate operators and the other is modified classical operators. By combining the two sets operators, the MDIPI model has the properties of the adaption and accuracy in hysteresis compensation. The MDIPI model can be used as feedforward controller to compensate different reference trajectories. Parameters of MDIPI model are calculated by matlab optimization tool box. The experiments of compensating the complex displacement trajectory and sinusoidal trajectory are validated on a platform of commercial PEA. The MDIPI model has achieved more accurate results than the Krasnosel’skii–Pokrovkii (KP), Preisach and CPI models. It is effective in improving the accuracy of PEA hysteresis compensation.

A Fractional-Order Normalized Bouc–Wen Model for Piezoelectric Hysteresis Nonlinearity

This article presents a new fractional-order normalized Bouc–Wen (BW) (FONBW) model to describe the asymmetric and rate-dependent hysteresis nonlinearity of piezoelectric actuators (PEAs). In view of the fact that the classical BW (CBW) model is only efficient for the symmetric and rate-independent hysteresis description, the FONBW model is devoted to characterizing the asymmetric and rate-dependent behaviors of the hysteresis in PEAs by adopting an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> th-order polynomial input function and two fractional operators, respectively. Different from the traditional modified BW models, the proposed FONBW model also eliminates the redundancy of parameters in the CBW model via the normalization processing. By this way, the developed FONBW model has a relatively simple mathematic expression with fewer parameters to simultaneously characterize the asymmetric and rate-dependent hysteresis behaviors of PEAs. Model parameters are identified by the self-adaptive differential evolution algorithm. To validate the effectiveness of the proposed model, a series of model verification and inverse-multiplicative-structure-based feedforward control experiments are carried out on a PEA system. Results show that the proposed model is superior to the CBW model and traditional modified BW model in modeling accuracy and hysteresis compensation.

Modeling and compensation for dynamic hysteresis of piezoelectric actuators based on Lissajous Curve

Piezoelectric actuators, which are widely used in micro-positioning systems, have hysteresis characteristics between the input and output. In particular, the hysteresis loop is complicated and difficult to be described when the input amplitude and frequency change. Most existing dynamic models cannot describe the law of dynamic hysteresis accurately, such as describing the zero drift of the hysteresis loop. In this paper, a novel and accurate model based on the Lissajous Curve is proposed to describe and compensate the dynamic hysteresis. Firstly, the performance of the piezoelectric actuator is tested, and the experimental results are quantified to research the law of dynamic hysteresis. Secondly, the Lissajous Curve Prandtl-Ishlinskii (LC-PI) model is proposed and the influence of differential order on the operator and memoryless function is discussed. Thirdly, three experiments are designed to verify the LC-PI model. Compared with the existing models, the LC-PI model can describe different types of dynamic hysteresis loops more accurately under the same modeling conditions. Finally, the LC-PI model is used to compensate the dynamic hysteresis. Experimental results show that the compensator based on this model can effectively suppress the dynamic hysteresis of the piezoelectric actuator.

Discrete-time rate-dependent hysteresis modeling and parameter identification of piezoelectric actuators

A discrete-time-modified Bouc–Wen model is proposed to describe the non-symmetrical and rate-dependent hysteresis of piezoelectric actuators for micro-vibration control applications. The modified model combines a non-symmetrical Bouc–Wen model and a frequency-dependent dynamic module. A series of experiments are conducted to characterize the rate-dependent hysteresis of piezoelectric stack actuators under sinusoidal excitations at a range of 1 to 20 Hz. The experimental results verify the validity of the modified model. The modified Bouc–Wen model increases the complexity of Bouc–Wen hysteresis nonlinear differential equation, which brings difficulties to parameter identification. To identify the parameters of Bouc–Wen model, an improved hybrid differential evolution and Jaya (DE-Jaya) algorithm is introduced with a hybrid mutant operator and Jaya operator that tried to balance between convergence speed and solution accuracy. The improved algorithm is tested on benchmark functions and compared with other optimizations to prove its effectiveness. The comparison results show that hybrid DE-Jaya algorithm has better performance in convergence speed and solution accuracy. The identified discrete-time-modified Bouc–Wen model is used as the secondary path in a filtered-x variable step-size affine projection algorithm (FXVSSAPA), and experimental verifications are done on a micro-vibration control platform. The experimental results show that the FXVSSAPA algorithm can converge to the steady-state error faster and verify the effectiveness of the proposed discrete-time-modified Bouc–Wen model.

Model Reference Adaptive Control of Cross-Coupling Hysteresis in Piezoceramics With Dynamic Loads

During the ultraprecise cutting of micro-structure surface with fast tool servo (FTS), the hysteresis of piezoelectric actuators (PEAs) are affected by dynamic voltage excitations and real-time cutting force, which declines the servo accuracy and cutting performance. In this paper, for a multi-input-single-output (MISO) cutting system, a cross-coupling rate-dependent Prandtl-Ishlinskii (CRPI) model is proposed and identified for the dynamic hysteresis of PEAs under dynamic voltage excitation and external loads. A model reference adaptive control method is then presented to eliminate the positioning nonlinearity of PEAs. The hysteresis modeling accuracy is discussed and the adaptive controller is validated through experiments.

Generalized regression neural network modeling based on inverse Duhem operator and adaptive sliding mode control for hysteresis in piezoelectric actuators

A dynamic generalized regression neural network model based on inverse Duhem operator is proposed to characterize the rate-dependent hysteresis in piezoelectric actuators. As hysteresis is multi-valued mapping, and traditional neural network can only model the system with one-to-one mapping. An inverse Duhem operator is proposed to extract the dynamic property of the hysteresis. Moreover, it can transform the multi-valued mapping of the hysteresis into a one-to-one mapping to suit the input of neural network. In order to compensate the effect of the hysteresis in piezoelectric actuator, the adaptive sliding mode controller with a feedforward hysteresis compensator is developed for the tracking control of the piezoelectric actuator. Experimental results demonstrate superior tracking performance, which validate the practicability and effectiveness of the presented approach.

Improved sliding mode dynamic matrix control strategy: Application on spindle loading and precision measuring device based on piezoelectric actuator

Spindle workload simulation and precision measurement are important in evaluating the precision of the spindle under different working conditions. Controlling the loading and measuring system accurately is difficult due to the robustness requirements of the controller, the complex dynamic model of the spindle loading system, the difficulty of obtaining system parameters, and the hysteresis of the piezoelectric actuator (PEA). To overcome this challenge, this study refers to the dynamic matrix control (DMC) method and proposes a highly versatile, robust, improved sliding mode DMC predictive controller (ISLDMC). ISLDMC is a non-inverse model control method with strong stickiness and good tracking characteristics. To help the controller achieve good versatility, this study regards nonlinear links, such as hysteresis of PEA, as an unknown disturbance and collects the step response signals of the piezoelectric system as the model state equation. In consideration of the practical application of the piezoelectric system, the prediction function of the state equation is used at the feedforward end of the controller, and the sliding mode function and incremental output are unified at the output end via weighted form optimization. In the feedback part, a variable correction coefficient is introduced into the state equation of the controller to enhance the tracking characteristics of the controller. This improvement is independently verified in a simulation environment. The design method of the controller is provided, the Schur stability of the closed-loop control system is proven. Moreover, the proposed controller is applied to machine tool spindle workload simulation and a precision-measuring device based on piezoelectric actuators. ISLDMC-based displacement-tracking and force-tracking controllers are designed. The experiment showed that the tracking performance and anti-interference, anti-model mismatch, and linear compensation capabilities of ISLDMC are better than those of the DMC controller, PI controller, and controllers from other studies. Furthermore, the linear compensation and anti-disturbance characteristics of the ISLDMC controller with the step response as the state equation are verified through force tracking experiments, which reveal that ISLDMC has good force tracking performance and anti-disturbance and linear compensation characteristics. The effectiveness of the introduction of dynamic correction coefficients is verified through a separate simulation involving displacement tracking and piezoelectric load experiments. The findings confirm that compared with the DMC controller, the ISLDMC controller improves tracking characteristics to a certain extent and has higher robustness.

Hysteresis Compensation for a Piezoelectric Actuator of Active Helicopter Rotor Using Compound Control.

Active rotor with trailing-edge flaps is a promising method to alleviate vibrations and noise level of helicopters. Hysteresis of the piezoelectric actuators used to drive the flaps can degrade the performance of an active rotor. In this study, bench-top tests are conducted to measure the nonlinear hysteresis of a double-acting piezoelectric actuator. Based on the experimental data, a rate-dependent hysteresis model is established by combining a Bouc–Wen model and a transfer function of a second order system. Good agreement is exhibited between the model outputs and the measured results for different frequencies. A compound control regime composed of a feedforward compensator and PID (Proportional–Integral–Derivative) feedback control is developed to suppress the hysteresis of this actuator. Bench-top test results demonstrate that this compound control regime is capable to suppress hysteresis at different frequencies from 10 Hz to 60 Hz, and errors between the desired actuator outputs and the measured outputs are reduced dramatically at different frequencies, revealing that this compound control regime has the potential to be implemented in an active helicopter rotor to suppress actuator hysteresis.

A Digitized Representation of the Modified Prandtl-Ishlinskii Hysteresis Model for Modeling and Compensating Piezoelectric Actuator Hysteresis.

Piezoelectric actuators are widely used in micromanipulation and miniature robots due to their rapid response and high repeatability. The piezoelectric actuators often have undesired hysteresis. The Prandtl–Ishlinskii (PI) hysteresis model is one of the most popular models for modeling and compensating the hysteresis behaviour. This paper presents an alternative digitized representation of the modified Prandtl–Ishlinskii with the dead-zone operators (MPI) hysteresis model to describe the asymmetric hysteresis behavior of piezoelectric actuators. Using a binary number with n digits to represent the classical Prandtl–Ishlinskii hysteresis model with n elementary operators, the inverse model can be easily constructed. A similar representation of the dead-zone operators is also described. With the proposed digitized representation, the model is more intuitive and the inversion calculation is avoided. An experiment with a piezoelectric stacked linear actuator is conducted to validate the proposed digitized MPI hysteresis model and it is shown that it has almost the same performance as compared to the classical representation.

Modeling and Compensation for Asymmetrical and Dynamic Hysteresis of Piezoelectric Actuators Using a Dynamic Delay Prandtl-Ishlinskii Model.

Piezoelectric actuators are widely used in micro- and nano-manufacturing and precision machining due to their superior performance. However, there are complex hysteresis nonlinear phenomena in piezoelectric actuators. In particular, the inherent hysteresis can be affected by the input frequency, and it sometimes exhibits asymmetrical characteristic. The existing dynamic hysteresis model is inaccurate in describing hysteresis of piezoelectric actuators at high frequency. In this paper, a Dynamic Delay Prandtl–Ishlinskii (DDPI) model is proposed to describe the asymmetrical and dynamic characteristics of piezoelectric actuators. First, the shape of the Delay Play operator is discussed under two delay coefficients. Then, the accuracy of the DDPI model is verified by experiments. Next, to compensate the asymmetrical and dynamic hysteresis, the compensator is designed based on the Inverse Dynamic Delay Prandtl–Ishlinskii (IDDPI) model. The effectiveness of the inverse compensator was verified by experiments. The results show that the DDPI model can accurately describe the asymmetrical and dynamic hysteresis, and the compensator can effectively suppress the hysteresis of the piezoelectric actuator. This research will be beneficial to extend the application of piezoelectric actuators.

Modeling and Inverse Compensation of Cross-Coupling Hysteresis in Piezoceramics under Multi-Input.

In the fast tool servo (FTS) system for microstructure surface cutting, the dynamic voltage hysteresis of piezoelectric actuators (PEAs) and the cutting force produced in the manufacturing affect the driving accuracy and the cutting performance. For a multi-input-single-output (MISO) cutting system, in this paper, a dynamic hysteresis model based on a rate-dependent Prandtl–Ishlinskii model is proposed. A backpropagation neural network (BPNN) is established to describe the cross-coupling effect between the applied voltage and external load. An inverse dynamic model is developed to compensate the nonlinearity of PEAs. The accuracy of the model and its inverse is discussed and the performance of the inverse feedforward compensator is validated through experiments.