A piezo driver for piezoelectric bimorph actuators in micro-positioning application

Abstract

Piezo driver for piezoelectric bimorph actuators is the key component for micro-positioning application. Hysteresis nonlinearity of piezoelectric bimorph actuators limits the performance of micro-positioning systems. A novel piezo driver was developed with Prandtl-Ishlinskii (P-I) model and correction network. The combination of P-I model and correction network can describe the rate-dependent hysteresis. Experiments were performed to validate the proposed piezo driver. The results show that the piezo driver can compensate the hysteresis of the piezoelectric bimorph actuator in the frequency range of 0.1 Hz to 10 Hz, and the accuracy of micro-positioning have been greatly improved.

Keywords

Piezoelectric bimorph actuator hysteresis compensation piezo driver

1. Introduction

Micro-positioning systems have been widely applied in micro-manipulation, nanotechnology and other precision engineering [1,2]. Due to the high displacement resolution and fast response, most of the micro-positioning systems are designed with piezoelectric actuators [3–5]. A high-performance piezo driver is the key component for micro-positioning system to drive piezoelectric actuators. Classical piezo driver can be considered as a linear amplifier, and the positioning accuracy is usually limited by the hysteresis [6–9].

To improve the accuracy of micro-positioning system, many methods have been developed to update the piezo driver [10–12]. In ref. [13], the rate-dependent Prandtl-Ishlinskii model was modified to improve the tracking performance of atomic force microscope scanner. In ref. [14], the Bouc-Wen model was proposed to describe the frequency-dependent behavior of piezoelectric actuators. The rate-dependent hysteresis of piezoelectric bimorph beam was analyzed in ref. [8].

In this paper, a piezo driver for micro-positioning application of piezoelectric bimorph was proposed with hybrid model. Since the hybrid model can describe the inverse of the hysteresis, the proposed piezo driver can compensation the hysteresis nonlinearity and improve the accuracy of micro-positioning.

2. Hybrid model of P-I model and correction network

To compensate the hysteresis in a wide frequency band, a hybrid model for piezoelectric bimorph actuator was developed with P-I model and correction network. The diagram of the proposed hybrid model is shown in Fig. 1, where the P-I model describe the hysteresis, and the correction network models the dynamics.

Fig. 1.

Diagram of the proposed hybrid model.

The P-I model is a phenomenological model whose basic hysteresis unit is Play operator. The expression of Play operator is shown in formula (1). $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}F_{r}[v](0)=f_{r}(v(0),0)=w(0)\\ F_{r}[v](t)=w(t)=f_{r}(v(t),F_{r}[v](t_{i}))\\ t_{i}<t\leq t_{i+1},0\leq i\leq N-1\\ f_{r}(v,w)=\max (v-r,\min ((v+r),w))\end{array} & & \displaystyle\end{eqnarray}$ (1) where r is the threshold, v is the input, Fr[v](t) is the output, and Fr[v](0) is the initial value.

The Play operators with different thresholds compose the P-I model, which is shown in formula (2). $\begin{eqnarray}\displaystyle x(t)={\Pi}[v](t)=qv(t)+\int _{0}^{R}p(r)F_{r}[v](t)dr & & \displaystyle\end{eqnarray}$ (2) where v is the input of P-I model, Fr[v](t) is the output of Play operator, p (r) is the weight of Play operator, q is the parameter and q > 0.

The dynamic response of piezoelectric bimorph actuators can be expressed as the sum of two second-order transfer functions [10]. In the micro-positioning application, the working frequency is far away from resonance. The dynamic response can be simplified and identified by measuring experiment, $\begin{eqnarray}\displaystyle G(s)=\frac{2400}{s^{2}+500s+80000}. & & \displaystyle\end{eqnarray}$ (3)

According to the control theory, the correction network can be expressed as $\begin{eqnarray}\displaystyle G^{\prime }(s)=\frac{1905s+266700}{s+8000}. & & \displaystyle\end{eqnarray}$ (4)

3. Circuit design of piezo driver

The circuit design of proposed piezo driver is shown in Fig. 2, which consists of boost circuit, full bridge invert circuit and DC-DC circuit.

Fig. 2.

Circuit diagram of piezo driver.

The TL494 is a PWM control chip, which can produce PWM signal and adjust PWM signal according to feedback voltage. The output voltage of boost circuit varies with the change of PWM signal. So that the output voltage can be stabilized at the design value. The IR2104S is a half-bridge driver, which can drive two MOSFET switches at the same time. The bridge invert circuit consists of four MOSFET switches. The computer outputs SPWM signal to control the turn-on order of the switches.

In order to compensate for the hysteresis effect of piezoelectric bimorph actuators, the P-I hysteresis model needs to be inverted. The parameters of P-I inverse model can be obtained by formula (5). $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\hat{q}={\displaystyle \frac{1}{q}}\\[8.0pt] \hat{r}_{j}=qr_{j}+\displaystyle \mathop{\sum }_{i=1}^{j-1}p_{i}(r_{j}-r_{i})\\ \hat{p}_{j}=-{\displaystyle \frac{p_{j}}{\left(q+\displaystyle \mathop{\sum }_{i=1}^{j}p_{i}\right)\left(q+\displaystyle \mathop{\sum }_{i=1}^{j-1}p_{i}\right)}}\end{array} & & \displaystyle\end{eqnarray}$ (5) where $\hat{q},\hat{r},\hat{p}$ are parameters of inverse model, q, r, p are parameters of P-I model.

In order to compensate the hysteresis at different frequencies, it is necessary to control the driving system. The diagram of driving power is shown in Fig. 3.

Fig. 3.

The diagram of piezo driver.

4. Results

To validate the proposed piezo driver, experiments were performed in different frequency. As shown in Fig. 4, the motion of piezoelectric bimorph actuator was measured by a laser sensor. P-I model with eight operators and correction network were implemented in CompactRIO. In the experiment, the threshold is r = [0.0625, 0.125, 0.1875, 0.25, 0.3125, 0.375, 0.4375, 0.5]. The parameters of P-I hysteresis model are identified from the experimental hysteresis data as shown in Table 1.

Fig. 4.

Experimental setup.

Table 1

The parameters of P-I model

Parameters	Value	r value
q	0.446	–
p ₁	0.113	0.0625
p ₂	0.3446	0.125
p ₃	−0.062	0.1875
p ₄	0.3069	0.25
p ₅	−0.053	0.3125
p ₆	0.0762	0.375
p ₇	0.4375	0.4375
p ₈	−0.637	0.5

Experiments were performed at 0.1 Hz, 1 Hz, 5 Hz and 10 Hz frequencies. The results shown in Fig. 5 indicate that the hysteresis error increases with the frequency, and the proposed piezo driver can effectively compensate the hysteresis, improve the positioning accuracy.

The hysteresis of piezoelectric bimorph actuator before compensation is very obviously, which is related to working frequency. The piezo driver includes inverse P-I model and correction network. The inverse P-I model compensates for the error caused by hysteresis and the correction network compensates for the error caused by dynamics. The error decreases obviously after compensation, which is shown in Fig. 6.

Fig. 5.

Major and minor hysteresis loops at 0.1 Hz, 1 Hz, 5 Hz and 10 Hz (Red curve: hysteresis curve after compensation; Blue curve: hysteresis curve before compensation).

Fig. 6.

Diagram of maximum error without and with compensation.

5. Conclusion

A piezo driver was developed with inverse P-I model and correction network. The inverse P-I model compensated hysteresis and the correction network solved dynamic delay. So the output of piezoelectric bimorph actuator can be controlled as we designed. This piezo driver can effectively compensate the hysteresis of piezoelectric bimorph actuator, improve the accuracy.

Footnotes

Acknowledgements

This is supported in part by National Natural Science Foundation of China (No. 51875277, 51607091).

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