Rate-dependent Hysteresis In Piezoelectric Actuators Research Articles

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.

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.

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.

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.

Elman neural network–based identification of rate-dependent hysteresis in piezoelectric actuators

Rate-dependent hysteresis nonlinearity in piezoelectric actuators severely limits micro- and nanoscale system performance. It is necessary to establish a dynamic model to describe the full behavior of rate-dependent hysteresis. In this article, the Elman neural network–based hysteresis model is developed for piezoelectric actuators. An improved dynamic hysteretic operator is proposed to transform the multi-valued mapping of hysteresis into one-to-one mapping on a newly constructed expanded input space. Then, Elman neural network incorporated with the improved dynamic hysteretic operator is utilized to approximate the behavior of rate-dependent hysteresis. The combination of Elman neural network and the improved dynamic hysteretic operator can dually embody the dynamic property and is capable of fully extracting the characteristics of rate-dependent hysteresis. The experimental results are presented to illustrate the potential of the proposed modeling technique.

Hysteresis modeling of piezoelectric actuators with the frequency-dependent behavior using a hybrid model

In order to overcome the problem of positioning inaccuracy caused by nonlinear hysteresis of piezoelectric actuators, a hybrid model based on least-squares support vector machine and Bouc–Wen model is proposed to model the rate-dependent hysteresis of piezoelectric actuators. A rate-independent Bouc–Wen model and its parameters identification method is established as the basis of least-squares Bouc–Wen model. Least-squares support vector machine, which is optimized by particle swarm optimization, is introduced to improve the Bouc–Wen model into a rate-dependent model by adjusting parameters of Bouc–Wen dynamically. Experiment is carried out to validate both Bouc–Wen model and the proposed least-squares Bouc–Wen model. The results show that the proposed least-squares Bouc–Wen method is a valid and a more precise method compared to the rate-independent Bouc–Wen model.

Fractional-order model and experimental verification for broadband hysteresis in piezoelectric actuators

Piezoelectric actuators are increasingly popular for high-accuracy and high-speed nanopositioning systems. However, nonlinear rate-dependent hysteresis of piezoelectric actuators leads to many difficulties in accurately analyzing the dynamic characteristics of piezoelectric nanopositioning systems in a wide frequency band. This paper proposes fractional-order model methods to characterize the hysteresis of piezoelectric actuators in time and frequency domains. Input voltage and output displacement of the piezoelectric actuator in time domain are employed to identify the parameters of the fractional-order model. Moreover, amplitude and phase errors are utilized to obtain the fractional model in frequency domain. Simulations and experiments are performed to validate the effectiveness of the proposed fractional-order model. The results show that the identified model in frequency domain is preferable in a wider frequency range from 1 to 200 Hz, and the maximum error is about 4.47%.

Modeling rate-dependent hysteresis in piezoelectric actuators using T-S fuzzy system based on expanded input space method

Piezoelectric actuators (PEAs) are favorable choices in the nanopositioning applications due to their high stiffness, fast response and ultrahigh precision. However, the undesired hysteresis nonlinearity in PEAs can dramatically degrade the positioning/tracking performance. In this paper, a Takagi-Sugeno (T-S) fuzzy system-based approach of identification for rate-dependent hysteresis is proposed. First, a novel hysteretic operator is proposed to describe the change tendency and extract the dynamic property of rate-dependent hysteresis. Then, the proposed hysteretic operator is added into the inputs to construct the expanded input space. Thus, it may transform the multi-valued mapping of hysteresis into one-to-one mapping, which enables the T-S fuzzy system to model the hysteresis. Finally, the experimental results are presented to illustrate the potential of the proposed modeling technique.

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.

Modeling of rate-dependent hysteresis in piezoelectric actuators based on a modified Prandtl-Ishlinskii model

Modeling of rate-dependent hysteresis in piezoelectric actuators based on a modified Prandtl-Ishlinskii model

A Modified Prandtl-Ishlinskii Model for Rate-dependent Hysteresis Nonlinearity Using mth-power Velocity Damping Mechanism

Hysteresis of piezoelectric actuators is rate-dependent at high frequencies, but most of the hysteresis models are rate-independent and cannot describe the rate-dependent hysteresis nonlinearity independently. In this paper, a modified Prandtl-Ishlinskii (P-I) model is proposed to characterize the rate-dependent hysteresis of piezoelectric actuators under sinusoidal excitation. This model is formulated by a mth-power velocity damping model in conjunction with the rate-independent P-I model. The parameter identification of this model is divided into two steps using different experimental data and algorithms. The particle swarm optimization is introduced first to identify the rate-independent parameters, and the nonlinear least square method is adopted afterwards to identify the rate-dependent parameters which are functions of the excitation frequency. Moreover, the proposed P-I model is developed to describe hysteresis nonlinearity under triangular excitation by introducing weighted functions, i.e., Λ i. Finally, the model results attained under the sinusoidal and triangular inputs at different frequencies are compared with the corresponding experimental data. The comparisons demonstrate that the proposed P-I model can well describe hysteresis nonlinearity under sinusoidal excitation up to 1,500 Hz and triangular excitation up to 250 Hz, respectively.

A simple fuzzy system for modelling of both rate-independent and rate-dependent hysteresis in piezoelectric actuators

In this paper, a novel fuzzy system based method for modelling both rate-independent and rate-dependent hysteresis in the piezoelectric actuator is proposed. First, the partial Takagi–Sugeno (T–S) fuzzy rule is designed. The antecedent structure of the fuzzy system is identified through uniform partition of its input variable. Then, the parameters of the consequent structure are optimized via the recursive least squares (RLS) algorithm. The modelling method is simple to implement and highly efficient to compute. Experimental results show that the proposed method is efficient to model both rate-independent and rate-dependent hysteresis. Based on the inverse of the developed model, feed-forward hysteresis compensation experiments at the frequencies of 50Hz and 100Hz are also conducted with the hysteresis effects being obviously reduced. The major contribution of this paper is that the inverse of the model can be analytically computed and the method can be applied to the case of real-time on-line modelling.

Modeling of rate-dependent hysteresis in piezoelectric actuators using a family of ellipses

Abstract In this paper, a new ellipse-like mathematic model is proposed to describe the rate-dependent hysteresis in piezoelectric actuators. Since the expressions of the model are completely analytical and can be determined only by a set of parameters, this method simplifies the modeling of complicated hysteresis behaviors. To represent the hysteresis effects, experiments are performed with designed sinusoidal excitations under different frequencies in the range 0.5–300 Hz. The rate-dependent hysteresis is characterized as increasing maximum hysteresis error (MHE) and decreasing peak-to-peak output amplitude (PPOA) phenomenons with the increase of input frequencies. Then, the parameters of the developed model are extracted from the experimental data using the direct least square method through MATLAB offline. The simulation results well correspond to the measured data and demonstrate that the developed model can precisely predict the rate-dependent hysteresis. We also investigate the parameters’ properties with hysteresis characteristics. In the developed model, the length of the minor radius describes the MHE varying with the input frequencies and amplitudes, while the length of major radius and the orientation of the ellipses represent the decreasing PPOA phenomenon. Finally, a real-time feedforward controller with an inverse model is designed to compensate for the rate-dependent hysteresis under different input frequencies. The experimental results show that the hysteresis effects are obviously reduced at both the lower and higher frequencies.

High-speed tracking control of piezoelectric actuators using an ellipse-based hysteresis model

In this paper, an ellipse-based mathematic model is developed to characterize the rate-dependent hysteresis in piezoelectric actuators. Based on the proposed model, an expanded input space is constructed to describe the multivalued hysteresis function H[u](t) by a multiple input single output (MISO) mapping Gamma:R(2)-->R. Subsequently, the inverse MISO mapping Gamma(-1)(H[u](t),H[u](t);u(t)) is proposed for real-time hysteresis compensation. In controller design, a hybrid control strategy combining a model-based feedforward controller and a proportional integral differential (PID) feedback loop is used for high-accuracy and high-speed tracking control of piezoelectric actuators. The real-time feedforward controller is developed to cancel the rate-dependent hysteresis based on the inverse hysteresis model, while the PID controller is used to compensate for the creep, modeling errors, and parameter uncertainties. Finally, experiments with and without hysteresis compensation are conducted and the experimental results are compared. The experimental results show that the hysteresis compensation in the feedforward path can reduce the hysteresis-caused error by up to 88% and the tracking performance of the hybrid controller is greatly improved in high-speed tracking control applications, e.g., the root-mean-square tracking error is reduced to only 0.34% of the displacement range under the input frequency of 100 Hz.

Neural networks based identification and compensation of rate-dependent hysteresis in piezoelectric actuators

Abstract This paper presents a method of the identification for the rate-dependent hysteresis in the piezoelectric actuator (PEA) by use of neural networks. In this method, a special hysteretic operator is constructed from the Prandtl–Ishlinskii (PI) model to extract the changing tendency of the static hysteresis. Then, an expanded input space is constructed by introducing the proposed hysteretic operator to transform the multi-valued mapping of the hysteresis into a one-to-one mapping. Thus, a feedforward neural network is applied to the approximation of the rate-independent hysteresis on the constructed expanded input space. Moreover, in order to describe the rate-dependent performance of the hysteresis, a special hybrid model, which is constructed by a linear auto-regressive exogenous input (ARX) sub-model preceded with the previously obtained neural network based rate-independent hysteresis sub-model, is proposed. For the compensation of the effect of the hysteresis in PEA, the PID feedback controller with a feedforward hysteresis compensator is developed for the tracking control of the PEA. Thus, a corresponding inverse model based on the proposed modeling method is developed for the feedforward hysteresis compensator. Finally, both simulations and experimental results on piezoelectric actuator are presented to verify the effectiveness of the proposed approach for the rate-dependent hysteresis.

A hybrid model for rate-dependent hysteresis in piezoelectric actuators

A hybrid model is proposed in this paper to describe the static nonlinear and dynamic characteristics of rate-dependent hysteresis in piezoelectric actuators. In this model, a neural network based submodel is implemented to approximate the static nonlinear characteristic of the hysteresis while a submodel with the first-order differential operators is constructed to describe the dynamic behavior of the hysteresis. In this paper, a special hysteretic operator is proposed to extract the hysteretic feature of the hysteresis. Then, an expanded input space with such special hysteretic operator is constructed. Based on the constructed expanded input space, the neural network can be implemented to approximate the hysteresis phenomenon. The submodel to describe the dynamics is a sum of the weighted first-order differential operators. Finally, the experimental results of applying the proposed method to the modeling of hysteresis in a piezoelectric actuator are also presented.

Modeling of hysteresis in piezoelectric actuators using neural networks

For the application of neural networks to the approximation of hysteresis which is characterized of multi-valued mapping and non-smooth nonlinearities, a novel modeling technique based on a transformation of one-to-one mapping is proposed in this paper. In this method, a special hysteretic operator is introduced to describe the change tendency of the hysteresis with regard to its input. Then an expanded input space is constructed for hysteresis with the introduction of such hysteretic operator, on which the multi-valued hysteresis is decomposed into a one-to-one mapping. Thus, neural networks model for hysteresis is derived, avoiding the calculation of the gradient of hysteresis. Subsequently, for approximation of rate-dependent hysteresis in piezoelectric actuators which is caused by the dynamic voltage excitations, a hybrid model, i.e. the dynamic extension of the proposed neural hysteresis submodel is developed. In the model, a linear dynamic block is introduced in series with the proposed neural model to allow for rate-dependent dynamics of the piezoelectric actuator simultaneously. Also the corresponding optimization algorithm by use of the modified Levenberg–Marquarqt (MLM) method is given. Finally, the experimental validation results of applying both the proposed neural hysteresis model and hybrid model to a piezoelectric actuator are presented.

Diagonal recurrent neural network with modified backlash operators for modeling of rate-dependent hysteresis in piezoelectric actuators

A novel neural network based method for modeling of rate-dependent hysteresis in piezoelectric actuators is proposed. In order to approximate the behavior of rate-dependent hysteresis which is a kind of nonsmooth dynamic nonlinearity with multi-valued mapping, a diagonal recurrent neural network (DRNN) with modified backlash operators (MBOs) is developed. In the proposed neural architecture, the MBOs are used as the activation functions of the hidden layer. Then, the corresponding Levenberg–Marquardt (L-M) algorithm for the DRNN with MBOs is proposed. Taking into account the nonsmooth characteristic of the MBO, the proximal bundle (PB) method is applied to search the appropriate subgradients at the nonsmooth vertexes of the MBOs. Finally, the experimental result on rate-dependent hysteresis in piezoelectric actuators is presented.