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Rate-dependent asymmetric hysteresis modeling and robust adaptive trajectory tracking for piezoelectric micropositioning stages

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Abstract

Hysteresis is an inherent characteristic of piezoelectric materials that can be determined by not only the historical input but also the input signal frequency. Hysteresis severely degrades the positioning precision of piezoelectric micropositioning stages. In this study, the hysteresis characteristics and the excitation frequency effects on the hysteresis behaviors of the piezoelectric micropositioning stage are investigated. Accordingly, a rate-dependent asymmetric hysteresis Prandtl–Ishlinskii (RDAPI) model is developed by introducing a dynamic envelope function into the play operators of the Prandtl–Ishlinskii (PI) model. The RDAPI model uses a relatively simple analytical structure with fewer parameters and then other modified PI models to characterize the rate-dependent and asymmetric hysteresis behavior in piezoelectric micropositioning stages. Considering practical situations with the uncertainties and external disturbances associated with the piezoelectric micropositioning stages, the system dynamics are described using a second-order differential equation. On this basis, a corresponding robust adaptive control method that does not involve the construction of a complex hysteretic inverse model is developed. The Lyapunov analysis method proves the stability of the entire closed-loop control system. Experiments confirm that the proposed RDAPI model achieves a significantly improved accuracy compared with the PI model. Furthermore, compared with the inverse RDAPI model-based feedforward compensation and the inverse RDAPI model-based proportional–integral–derivative control methods, the proposed robust adaptive control strategy exhibits improved tracking performance.

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Data statement

The datasets generated during the current study are available from the corresponding author on reasonable request.

Abbreviations

\(b_{s}\) :

Damping coefficient

\(C_{A}\) :

Sum of the capacitance of the total piezoelectric ceramics

d :

External disturbance

\(e_{1}\) :

Tracking error

\(e_{2}\) :

Intermediate variable

F :

Fitness

\(F_{A}\) :

Transduced force from the electrical side

\(F_{r} {[u]}(t)\) :

Classical PI play operator

\(F^{h}_{r}{[u]}(t)\) :

Proposed RDAPI play operator

H :

Hysteresis effect

\(H'\) :

Hysteresis nonlinearity

\(k_{{\textit{amp}}}\) :

Fixed gain of the voltage power amplifier

\(k_{i}\), \(i=1,2\) :

Positive design parameter

\(k_{s}\) :

Moving mechanism stiffness

m :

Moving mechanism mass

M :

Population size

\(p_{i}\), \(i=1,..,n\) :

Density parameter

p, q, and n :

Unknown system parameters

q :

Total charge in the PCA

\(\dot{q}\) :

Resulting current flowing through the circuit

\(q_{c}\) :

Charge stored in the linear capacitance \(C_{A}\)

\(q_{p}\) :

Transduced charge from the mechanical side due to the piezoelectric effect

r :

Threshold of play operator

\(R_{0}\) :

Equivalent internal resistance of driving circuit

\(T_{em}\) :

Electromechanical transducer with transformer ratio

u :

Input of the piezoelectric driver

\(v_{A}\) :

Transduced voltage

\(v_{h}\) :

Generated voltage due to H

\(w^{t}_{1}\) and \(w^{t}_{1}\) :

Chromosome

\(x_{1}=x\) :

Output displacement

\(x_{2}\) :

Time derivative of \(x_{1}\)

\(x_{d}\) :

Reference trajectory

\(\alpha ,\beta \) :

RDAPI model parameters

\(\lambda \), \(\gamma \), and \(\eta \) :

Adjusting parameters of adaptive update laws

\(\rho \) :

Boundary of the disturbance d(t), \(d(t)\le \rho \)

\(\varsigma \) :

Boundary of the nonlinear part \( \alpha {{u}^{2}}\left( t\right) +\sum _{i=1}^{n}{{{p}_{i}}F_{r}^{h}\left[ u\right] \left( t\right) }\)

\(\omega \) :

Unknown hysteresis nonlinearity

AME:

Absolute mean error

PCA:

Piezoelectric ceramic actuator

PI:

Prandtl–Ishlinskii model

RDAPI:

Rate-dependent asymmetric PI model

RMSE:

Root-mean-square error values

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grant 51675228 and the Program of Science and Technology Development Plan of Jilin Province of China under Grants 20180101052JC and 20190303020SF.

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Authors and Affiliations

  1. Department of Control Science and Engineering, Jilin University, Changchun, 130022, China

    Linlin Nie, Yiling Luo, Wei Gao & Miaolei Zhou

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Nie, L., Luo, Y., Gao, W. et al. Rate-dependent asymmetric hysteresis modeling and robust adaptive trajectory tracking for piezoelectric micropositioning stages. Nonlinear Dyn 108, 2023–2043 (2022). https://doi.org/10.1007/s11071-022-07324-7

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  • DOI: https://doi.org/10.1007/s11071-022-07324-7

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