Hysteresis Compensation and Butterworth Pattern-Based Positive Acceleration Velocity Position Feedback Damping Control of a Tip-Tilt-Piston Piezoelectric Stage

Abstract

In order to solve the hysteresis nonlinearity and resonance problems of piezoelectric stages, this paper takes a three-degree-of-freedom tip-tilt-piston piezoelectric stage as the object, compensates for the hysteresis nonlinearity through inverse hysteresis model feedforward control, and then combines the composite control method of positive acceleration velocity position feedback damping control and high-gain integral feedback controller to suppress the resonance of the system and improve the tracking speed and positioning accuracy. Firstly, the three-degree-of-freedom motion of the end-pose is converted into the output of three sets of piezoelectric actuators and single-axis control is performed. Then, the rate-dependent Prandtl–Ishlinskii model is established and the parameters of the inverse model are identified. The accuracy and effectiveness of parameter identification are verified through open-loop and closed-loop compensation experiments. After that, for the third-order system, the parameters of positive acceleration velocity position feedback damping control and high-gain integral feedback controller are designed as a whole based on the pattern of the Butterworth filter. The effectiveness of the design method is proved by step signal and triangle wave signal trajectory tracking experiments, which suppresses the resonance of the system and improves the bandwidth of the system and the tracking speed of the stage.

1. Introduction

Due to the advantages of piezoelectric actuators (PEAs), such as fast response, large load capacity, high resolution and small size [1], piezoelectric-driven nanopositioning stages are currently widely used in ultra-precise machining, atomic force microscopy and fast steering mirrors [2]. However, since PEAs also have nonlinear characteristics, such as hysteresis and creep, they seriously affect the positioning accuracy and response speed of the positioning stage [3]. In addition, the inherent low damping characteristics of the piezoelectric-driven stage lead to a low gain margin of the system, which further affects the control bandwidth and stability of the stage [4]. Especially in high-frequency operation, the resonant characteristics caused by the low damping characteristics may even cause system instability [5]. Therefore, considering the compensation of the hysteresis nonlinearity of PEAs and the resonant control of stage is the key to improving the tracking accuracy and tracking speed of the piezoelectric-driven nanopositioning stage.

At present, the common method to compensate for the hysteresis nonlinearity of PEAs is to establish an accurate inverse hysteresis model for feedforward compensation [6]. In recent years, many models have been developed to accurately characterize the hysteresis nonlinearity, such as Duhem model [7], Preisach model [8], Prandtl–Ishlinskii (PI) model [9], Bouc–Wen model [10], etc. In addition, in order to further describe the rate-dependent characteristics of hysteresis, the Hammerstein model [11], rate-dependent fractional-order Duhem model [12], and rate-dependent PI model have also been proposed [13]. Among the above methods, the rate-dependent PI model has the advantages of reversibility and convenient real-time control, so this paper selects this model to describe the rate-dependent hysteresis characteristics of PEAs.

In order to solve the resonance problem caused by the low damping characteristics of the stage, the common method is to use feedback damping control, such as integral resonance control [14], force feedback control, positive position feedback (PPF) control [15], positive velocity position feedback (PVPF) control [16], positive acceleration velocity position feedback (PAVPF) control [17], etc. These methods can effectively improve the damping of the system and ensure the robustness of the system by changing the zero-pole distribution of the control system. Among these feedback damping control methods, PVPF control and PAVPF control have received more and more attention because they can realize the free pole configuration of the closed-loop system. However, PVPF control is generally designed for second-order systems, while the dynamic model of PEAs is generally modeled as a third-order system. Therefore, PAVPF designed for the third-order dynamic model is more suitable for controlling PEAs.

In addition, in order to improve the tracking accuracy of the stage, the common control method is to use dual closed-loop control. The inner loop uses a damping controller to suppress the resonance characteristics of the stage, and the outer loop uses a high-gain tracking controller to improve the tracking speed. However, the parameters of the dual loops are usually designed separately, which will cause the bandwidth of the system to be severely limited. At the same time, adjusting the gain of the tracking controller will also change the closed-loop damping of the system. Therefore, designing the parameters of the damping controller and the tracking controller at the same time can more effectively and scientifically improve the bandwidth of the system. Regarding the configuration method of the desired poles, Eieslen et al. described the closed-loop system after damping control in the pattern of a Butterworth filter to obtain the parameters of both the damping controller and the tracking controller, which provides a higher system bandwidth and smaller tracking error compared to traditional damping control [18], providing a new idea for this paper. Russell et al. have designed the parameters of IRC, PPF, and PVPF for the second-order system by this method and confirmed its effectiveness [19].

According to the above research status, focusing on the hysteresis nonlinearity of PEAs and the resonant characteristics of the stage, the main contribution of this paper is to first eliminate the influence of nonlinearity through inverse rate-dependent PI model feedforward compensation. Then, different from existing studies that model PEAs as second-order systems, for the third-order dynamic systems of PEAs, the PAVPF damping controller is designed to suppress the resonance of the system, and the Butterworth filter pattern is used to configure the poles of the closed-loop system as a whole. The parameters of the integral tracking controller and the PAVPF damping controller are designed together to further improve the damping of the system and the tracking speed and accuracy of the stage.

2. Experimental Environment

2.1. Experimental System

The experimental system is shown in Figure 1. The experimental stage of this paper is a tip-tilt-piston piezoelectric stage (PT4V150-400S-S) produced by Shanghai NanoMotions. The stage is driven by three sets of parallel PEAs which distributed at 120° to each other. Its Z-axis travel range is 0–200 µm, and the maximum deflection angles of the X-axis and Y-axis are ±2.2 mrad and ±2.5 mrad, respectively. The positioning stage control system is an xPC Target semi-physical real-time simulation system based on the host-target mode, which can see the running results of the control program in real time, facilitate the real-time adjustment of the controller and model parameters, and achieve ideal experimental results. The input voltage (0–120 V) of PEAs are obtained by amplifying the control signal (0–10 V) that from the data acquisition card (PCI-6221) installed on the target machine and inputing to the voltage amplifier (PCM931S). Each PEA is integrated with a resistive strain gauge sensor to obtain the output displacement of the PEA, and converted into a voltage signal (0–10 V) input to the data acquisition card with a resolution of 7 nm. The sampling frequency of experiments in this paper is 5 kHz.

The three degrees of freedom (3-DOF) of the stage are the Z-axis displacement $\Delta z$ and the deflection angles ${\theta }_{\mathrm{x}}$ and ${\theta }_{\mathrm{y}}$ around the X-axis and Y-axis, respectively, which are driven by three sets of PEAs. Since the three sets of actuators are parallel and relatively independent, in order to control conveniently, this paper converts the 3-DOF movement into the output of three sets of PEAs and performs single-axis control. The kinematic model of the stage is shown in Figure 2. The global coordinate O-$XYZ$ is fixed at the position shown in Figure 2. $P_1$, $P_2$, and $P_3$ are the output points of the three sets of piezoelectric drive units, which form an isosceles triangle with l = 89 mm and h = 77 mm. ${P^{\prime }}_4$ is the midpoint of ${P^{\prime }}_2$${P^{\prime }}_3$, and O is the midpoint of ${P^{\prime }}_1$${P^{\prime }}_4$. Before applying voltage, the positions of PEAs $P_1$, $P_2$, and $P_3$ are consistent with ${P^{\prime }}_1$, ${P^{\prime }}_2$, and ${P^{\prime }}_3$ on the plane $\mathsf{\Psi }$. After applying voltage, the three PEAs produce displacements $p_1$ = $P_1$${P^{\prime }}_1$, $p_2$ = $P_2$${P^{\prime }}_2$, $p_3$ = $P_3$${P^{\prime }}_3$, respectively. The angles ${\theta }_{\mathrm{x}}$ and ${\theta }_{\mathrm{y}}$, as shown in Figure 2, are positive. From the geometric model shown in Figure 2, we can obtain the following:

$$\begin{array}{cc}\hfill \Delta z& ={\displaystyle \frac{1}{2}}\left[p_1+{\displaystyle \frac{1}{2}}\left(p_2+p_3\right)\right],\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill {\theta }_{\mathrm{x}}& \approx tan{\theta }_{\mathrm{x}}={\displaystyle \frac{1}{l}}\left(p_3-p_2\right),\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\theta }_{\mathrm{y}}& \approx tan{\theta }_{\mathrm{y}}={\displaystyle \frac{1}{h}}\left[p_1-{\displaystyle \frac{1}{2}}\left(p_2+p_3\right)\right].\hfill \end{array}$$

(3)

The kinematic relationship between the output displacements of three sets of PEAs and 3-DOF motion is as follows:

$$\left[\begin{array}{c}{p}_1\\ p_2\\ p_3\end{array}\right]=\left[\begin{array}{ccc}{\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{4}}& {\displaystyle \frac{1}{4}}\\ 0& {\displaystyle \frac{1}{l}}& {\displaystyle \frac{1}{l}}\\ {\displaystyle \frac{1}{h}}& -{\displaystyle \frac{1}{2h}}& -{\displaystyle \frac{1}{2h}}\end{array}\right]\left[\begin{array}{c}\Delta z\\ {\theta }_{\mathrm{x}}\\ {\theta }_{\mathrm{y}}\end{array}\right].$$

(4)

2.2. Hysteresis and Resonance Characteristics of Stage

In order to obtain the hysteresis characteristics of PEAs, taking PEA1 as an example, a sinusoidal signal with an amplitude of 2 V and different frequencies is applied to the voltage amplifier, and then it is input into PEA1. The result is shown in Figure 3a. It can be seen that there is an obvious hysteresis loop relationship between the output displacement and the input signal. At the same time, as the frequency of the input signal increases, the hysteresis loop becomes wider. It can be seen that the hysteresis of the PEA has a certain rate-dependent characteristic, which needs to be taken into account when modeling. Then, in order to obtain the resonant characteristics of the positioning stage, a step signal with an amplitude of 1V is input to the PEA1 in open-loop, and the result is shown in Figure 3b. It can be seen that the positioning stage exhibits a strong resonant vibration characteristic when tracking high-frequency signals, which will have a great impact on the positioning accuracy and stability of the stage.

3. Hysteresis Modeling and Compensation

In this section, we mainly model and compensate for the hysteresis characteristics of PEAs. As mentioned above, hysteresis nonlinearity will seriously affect the positioning accuracy of the nanopositioning stage, and hysteresis also has certain rate-dependent characteristics. Therefore, this paper uses the rate-dependent PI hysteresis model which is easy to identify and apply to describe the rate-dependent hysteresis characteristics of PEAs and adopts inverse model feedforward compensation (IMFC).

3.1. Rate-Dependent PI Hysteresis Model

Firstly, in order to more accurately compensate for the hysteresis nonlinearity, the PI model, which is convenient for constructing the inverse hysteresis model, is chosen to characterize the hysteresis effect of the stage. The rate-dependent PI hysteresis model [20] is expressed as follows:

$$y_{\mathrm{m}}\left(t\right)=c_1{u}^3\left(t\right)+c_{2}u\left(t\right)+\sum _{i=1}^N{d}_i{F}_{r_i}^{\sigma }\left[u\right]\left(t\right),$$

(5)

where $u\left(t\right)$ is the input signal of the model, $y_{\mathfrak{m}}\left(t\right)$ is the output signal of the model, N is the number of the Play operators, which is set to 10 in this paper, and $F_{r_i}^{\sigma }\left[u\right]\left(t\right)$ is the output of the Play operator, expressed as

$$\left\{\begin{array}{c}{F}_{r_i}^{\sigma }\left[u\right]\left(t\right)=f_{r_i}^{\sigma }\left(u\left(t\right),y_{\sigma }(t-T)\right)=max(k_1,k_2)\hfill \\ k_1={\sigma }_1\left(u\left(t\right),\dot{u}\left(t\right)\right)-r_i\hfill \\ k_2=min\left({\sigma }_2\left(u\left(t\right),\dot{u}\left(t\right)\right),y_{\sigma }(t-T)\right)\hfill \end{array}\right.,$$

(6)

$$r_i={\displaystyle \frac{i-1}{N}}{∥\begin{array}{c}u\left(t\right)\end{array}∥}_{\infty },i=1,2,...,N,$$

(7)

$${\sigma }_1\left(u\left(t\right),\dot{u}\left(t\right)\right)=u\left(t\right)-c_3\left|\dot{u}\left(t\right)\right|,$$

(8)

$${\sigma }_2\left(u\left(t\right),\dot{u}\left(t\right)\right)=u\left(t\right)+c_4\left|\dot{u}\left(t\right)\right|,$$

(9)

$$\dot{u}\left(t\right)={\displaystyle \frac{u\left(t\right)-u(t-T)}{T}},$$

(10)

where T = 0.0002 s is the sampling period, $r_i$ and $d_i$ are the threshold and weight value of the Play operator, respectively, and ${\sigma }_1(u\left(t\right),\dot{u}\left(t\right))$ and ${\sigma }_2(u\left(t\right),\dot{u}\left(t\right))$ are the dynamic envelope function of the input function $u\left(t\right)$ and its derivatives $\dot{u}\left(t\right)$, respectively. $c_1$, $c_2$, $c_3$, $c_4$ are constants and are obtained together with $d_i$ from experimental data identification, where $c_3>0$, $c_4>0$, and $d_i>0$ in the positive model, and $d_i<0$ in the inverse model.

3.2. Hysteresis Model Identification

This paper adopts the IMFC control method to compensate for hysteresis nonlinearity, so only the parameters of the inverse model need to be identified. One of the advantages of the rate-dependent PI model is its reversibility. The parameters of the inverse model can be identified by simply exchanging the output signal with the input signal during identification. Assuming that the input signal $u\left(t\right)$ of the hysteresis model is the output of the inverse hysteresis model, and $y_{\mathrm{m}}\left(t\right)$ is the input of the inverse hysteresis model, the inverse PI model is expressed as

$$u\left(t\right)=\overline{c_1}{y}_{\mathrm{m}}^3\left(t\right)+\overline{c_2}{y}_{\mathrm{m}}\left(t\right)+\sum _{i=1}^N{\overline{d}}_i{F}_{\overline{r_i}}^{\sigma }\left[y_{\mathrm{m}}\right]\left(t\right),$$

(11)

where ${\overline{c}}_1$, ${\overline{c}}_2$, ${\overline{d}}_i$, and ${\overline{r}}_i$ are the parameters, weights and thresholds of the inverse hysteresis model, respectively. The system has three sets of PEAs corresponding to three independent PI models, so three different sets of model parameters need to be identified separately. Taking PEA1 as an example, considering the rate-dependent characteristic of the model, the identification signal should be the superposition signal of signals with different frequencies. Therefore, in the open-loop system, the input signal of PEA is as follows:

$$u\left(t\right)=5-2.5cos\left(3\pi t\right)-2.5cos\left(\pi t\right).$$

(12)

At the same time, the voltage signal (0.05 V/µm) corresponding to the actual output displacement of the PEA is obtained by the built-in sensor. It should be noted that since the output signal contains noise signals, the derivative will fluctuate violently when it is used as an input signal, affecting the identification accuracy. Therefore, it is necessary to use a Gaussian filter to smooth the output signal before identification, which can improve the accuracy of identification results. However, there is an error between the filtered signal and the ideal actual signal without noise, which leads to potential errors in the identification results of the model. The identification method entails using the Optimization toolbox in Matlab, importing the input signal and the output signal, and then using the constrained quadratic Optimization method of the following objective function for parameter identification:

$${\mathbf{\Phi }}_1\left(\mathbf{X}\right)=min\left\{{\left[{\mathbf{M}}_1-{\mathbf{U}}_1\right]}^{\mathrm{T}}\left[{\mathbf{M}}_1-{\mathbf{U}}_1\right]\right\},$$

(13)

where ${\mathbf{M}}_1={[y_{\mathrm{m}}\left(0\right),y_{\mathrm{m}}\left(1\right),\dots ,y_{\mathrm{m}}\left(n\right)]}^{\mathrm{T}}$ is the output displacement of the model, $n=t_{\mathrm{r}}/T_{\mathrm{s}}$, $t_{\mathrm{r}}$ is the time length of the input signal, $T_{\mathrm{s}}$ is the system sampling period, ${\mathbf{U}}_1={[y_{\mathrm{r}}\left(0\right),y_{\mathrm{r}}\left(1\right),\dots ,y_{\mathrm{r}}\left(n\right)]}^{\mathrm{T}}$ is the voltage signal converted from the actual displacement voltage signal of the PEA1 collected by the sensor, and $\mathbf{X}=[c_1,c_2,d_1,d_2,...,d_N,c_3,c_4]$ are all the parameters that need to be identified in the model. The inverse model parameters of PEA2 and PEA3 can also be identified by this method. The identification parameters of PEAs are shown in

Table 1. Parameter identification results of inverse PI models.

	${\mathbf{PI}}_1^{-1}$	${\mathbf{PI}}_2^{-1}$	${\mathbf{PI}}_3^{-1}$
$c_1$	0.0020	0.0021	0.0018
$c_2$	1.2910	1.2162	1.2381
$d_1$	−0.1365	0	0
$d_2$	−0.1501	−0.2819	−0.2889
$d_3$	−0.1298	−0.0447	−0.0359
$d_4$	−0.0586	−0.0847	−0.1055
$d_5$	−0.0097	−0.0283	−0.0234
$d_6$	−0.0239	0	0
$d_7$	−0.0209	−0.0002	0
$d_8$	0	−0.0001	0
$d_9$	−0.0053	−0.0001	0
$d_{10}$	−0.2098	−0.3307	−0.3814
$c_3$	0.0068	0.0042	0.0019
$c_4$	0.0073	0.0109	0.0091

, and the identification results are shown in Figure 4. It can be seen that the maximum identification error is only 3.8682 µm, and the root mean square error (RMSE) values of the identification results are 1.0014 µm, 0.9576 µm, 1.0327 µm, which can prove the accuracy of the identification results.

3.3. Inverse Hysteresis Model Feedforward Compensation

In order to verify the accuracy of inverse model identification results, this paper conducts open-loop and closed-loop IMFC experiments, respectively, and the control block diagram is shown in Figure 5. Firstly, in the open-loop experiment, the stage is allowed to track the increasing triangle wave signal, and the experimental results are shown in Figure 6a. It can be seen that the maximum width of the hysteresis loop is reduced by about 95%, and the hysteresis nonlinearity is well compensated, which proves the accuracy of inverse model identification results. Then, in the closed-loop experiment, the stage is allowed to track a sine wave signal, and the signal with an amplitude of 2 V and a frequency of 2Hz is applied to the voltage amplifier. The experimental results are shown in Figure 6b. It can be seen that only PID closed-loop control cannot reduce the influence of hysteresis nonlinearity. Through the composite control of IMFC and PID feedback control, the hysteresis loop is reduced by about 90%, which proves the effectiveness of the control method and the accuracy of inverse model identification results.

4. PAVPF Damping Control

The resonant vibration caused by the low damping characteristics of the piezoelectric driven nanopositioning stage will seriously affect the tracking accuracy and stability of the stage. In order to solve this problem, this section uses the pole configuration method based on the Butterworth filter pattern to simultaneously design the parameters of the PAVPF damping controller and the tracking controller together to increase the system damping, reduce the resonance effect, and improve the tracking accuracy of the system.

4.1. System Dynamics Model Identification

In order to obtain accurate dynamic characteristics, it is necessary to first identify the dynamic model of the positioning stage. Usually, the dynamic model of PEAs is modeled as a third-order two-zero transfer function. In order to identify the model, input band-limited white noise signals that can fully stimulate the dynamic characteristics of the system to PEAs and collect the output signals from the sensors. Then, the input signal and the output signal are imported into the Ident toolbox in Matlab for system identification. The identified system transfer function is shown in Equation (14). The comparison between the identification results and the actual frequency responses is shown in Figure 7. It can be seen that the accuracy of the identification result is very high and can characterize the dynamic characteristics of the system. At the same time, due to the inherent characteristics of the flexible mechanism, the stage has multiple resonant frequencies. In this paper, the first resonant frequency is taken as the main resonant frequency, and the remaining resonant effects are equivalent to system interference.

$$\left\{\begin{array}{c}{G}_{11}\left(s\right)={\displaystyle \frac{682.2s^2+2.43e5s+1.785e9}{s^3+1371s^2+1.722e6s+2.225e9}}\hfill \\ G_{22}\left(s\right)={\displaystyle \frac{789.3s^2+3.599e5s+1.495e9}{s^3+1398s^2+1.581e6s+1.953e9}}\hfill \\ G_{33}\left(s\right)={\displaystyle \frac{827.4s^2+2.15e5s+1.622e9}{s^3+1364s^2+1.537e6s+1.988e9}}\hfill \end{array}\right..$$

(14)

In order to facilitate parameter design, the system needs to be simplified into a third-order system without zero points. It is only necessary to ensure that the simplified system has the same damping ratio, resonant frequency, and direct gain as the original system. The simplified transfer function is shown in Equation (15). The simplified transfer function parameters of the three sets of PEAs are shown in

Table 2. Simplified transfer function parameters.

	${\mathit{G}}_1\left(\mathit{s}\right)$	${\mathit{G}}_2\left(\mathit{s}\right)$	${\mathit{G}}_3\left(\mathit{s}\right)$
$\mathsf{\Gamma }$	1.785 × 109	1.495 × 109	1.622 × 109
$b_2$	1371	1398	1364
$b_1$	1.722 × 106	1.581 × 106	1.537 × 106
$b_0$	2.225 × 109	1.953 × 109	1.988 × 109

.

$$G\left(s\right)={\displaystyle \frac{\mathsf{\Gamma }}{s^3+b_2{s}^2+b_{1}s+b_0}}.$$

(15)

4.2. PAVPF Parameter Design

In order to suppress the resonant characteristics of the system, this paper adopts the PAVPF damping controller $C_{\mathrm{d}}\left(s\right)$. At the same time, in order to improve the tracking accuracy, the integral tracking controller $C_{\mathrm{t}}\left(s\right)$ is used as the feedback controller. The control block diagram of the system is shown in Figure 8.

$C_{\mathrm{d}}\left(s\right)$ and $C_{\mathrm{t}}\left(s\right)$ are expressed as follows:

$$\begin{array}{cc}\hfill C_{\mathrm{d}}\left(s\right)=& {\displaystyle \frac{{\lambda }_2{s}^2+{\lambda }_{1}s+{\lambda }_0}{s^2+2{\xi }_{\mathrm{d}}{w}_{\mathrm{d}}s+w_{\mathrm{d}}^2}},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill C_{\mathrm{t}}\left(s\right)=& {\displaystyle \frac{k_{\mathrm{t}}}{s}}.\hfill \end{array}$$

(17)

Among them, ${\lambda }_2$, ${\lambda }_1$, ${\lambda }_0$, ${\xi }_{\mathrm{d}}$, $w_{\mathrm{d}}$, and $k_{\mathrm{t}}$ are all parameters that need to be designed. In fact, $C_{\mathrm{d}}\left(s\right)*u\left(t\right)$ can be regarded as a low-pass filter multiplied by the position, velocity and acceleration of the input signal. Since the sensor can only collect the output position signal, the velocity signal and acceleration signal are obtained by the first and second differential of the position signal. ${\lambda }_2$, ${\lambda }_1$, and ${\lambda }_0$ are the weight values of the acceleration signal, velocity signal and position signal, respectively. At the same time, the low-pass filter can avoid amplifying sensor noise during differentiation. One of the advantages of using the PAVPF damping controller is that the poles can be configured freely. This paper uses the Butterworth filter pattern to configure the poles. The specific process is as follows.

Firstly, as shown in Figure 8, the closed-loop transfer function of the system can be calculated as

$$G_{\mathrm{sys}}\left(s\right)={\displaystyle \frac{C_{\mathrm{d}}\left(s\right)G_{\mathrm{l}}\left(s\right)}{1-G_{\mathrm{l}}\left(s\right)(C_{\mathrm{d}}\left(s\right)-C_{\mathrm{t}}\left(s\right))}}={\displaystyle \frac{N\left(s\right)}{D\left(s\right)}},$$

(18)

where

$$N\left(s\right)=k_{\mathrm{t}}\mathsf{\Gamma }(s^2+2{\xi }_{\mathrm{n}}{w}_{\mathrm{n}}s+w_{\mathrm{n}}^2),$$

(19)

$$\begin{array}{cc}\hfill D\left(s\right)=& s^6+(b_2+2{\xi }_{\mathrm{d}}{w}_{\mathrm{d}})s^5+(w_{\mathrm{d}}^2+b_2·2{\xi }_{\mathrm{d}}{w}_{\mathrm{d}}+b_{\mathrm{l}})s^4\hfill \\ \hfill & +(b_2·w_{\mathrm{d}}^2+b_1·2{\xi }_{\mathrm{d}}{w}_{\mathrm{d}}+b_0-\mathsf{\Gamma }·{\lambda }_2)s^3\hfill \\ \hfill & +(b_{\mathrm{l}}·w_{\mathrm{d}}^2+b_0·2{\xi }_{\mathrm{d}}{w}_{\mathrm{d}}-\mathsf{\Gamma }·{\lambda }_{\mathrm{l}}+k_{\mathrm{t}}·\mathsf{\Gamma })s^2\hfill \\ \hfill & +(b_0·w_{\mathrm{d}}^2-\mathsf{\Gamma }·{\lambda }_0+k_{\mathrm{t}}\mathsf{\Gamma }·2{\xi }_{\mathrm{d}}{w}_{\mathrm{d}})s+k_{\mathrm{t}}\mathsf{\Gamma }{w}_{\mathrm{d}}^2.\hfill \end{array}$$

(20)

Then, the corresponding characteristic equation of the ideal sixth-order Butterworth filter is

$$\begin{array}{cc}\hfill P\left(s\right)=& (s^2+2{\xi }_1{w}_{\mathrm{b}}s+w_{\mathrm{b}}^2)(s^2+2{\xi }_2{w}_{\mathrm{b}}s+w_{\mathrm{b}}^2)(s^2+2{\xi }_3{w}_{\mathrm{b}}s+w_{\mathrm{b}}^2)\hfill \\ \hfill =& s^6+(2{\xi }_1+2{\xi }_2+2{\xi }_3)w_{\mathrm{b}}{s}^5+(3+4{\xi }_1{\xi }_2+4{\xi }_1{\xi }_3+4{\xi }_2{\xi }_3)w_{\mathrm{b}}^2{s}^4\hfill \\ \hfill & +(4{\xi }_1+4{\xi }_2+4{\xi }_3+8{\xi }_1{\xi }_2{\xi }_3)w_{\mathrm{b}}^3{s}^3+(3+4{\xi }_1{\xi }_2+4{\xi }_1{\xi }_3+4{\xi }_2{\xi }_3)w_{\mathrm{b}}^4{s}^2\hfill \\ \hfill & +(2{\xi }_1+2{\xi }_2+2{\xi }_3)w_{\mathrm{b}}^{5}s+w_{\mathrm{b}}^6\hfill \\ \hfill =& s^6+K_5{s}^5+K_4{s}^4+K_3{s}^3+K_2{s}^2+K_{1}s+K_0,\hfill \end{array}$$

(21)

where ${\xi }_1$, ${\xi }_2$, ${\xi }_3$ are the desired closed-loop damping ratios, ${\xi }_1$ = $cos(\pm {15}^{\circ })$ = $0.9659$, ${\xi }_2$ = $cos(\pm {45}^{\circ })$ = 0.7071, ${\xi }_3$=$cos(\pm {75}^{\circ })$ = $0.2588$, and the desired natural frequency $w_{\mathrm{b}}$ can be freely set. Under the condition of ensuring stability, the desired natural frequency is as close to the first resonant frequency as possible. In this paper, the $w_{\mathrm{b}}$ of the three sets of PEAs are set as 1120, 1080, and 1090, respectively. The pole distribution of the ideal closed-loop system in the Butterworth filter pattern is shown in Figure 9. They are evenly distributed in the left half plane. Each pole of the ideal sixth-order system is ${30}^{\circ }$ apart. By equating Equation (14) with Equation (15), the following relationship can be derived:

$$\begin{array}{cc}\hfill 2{\xi }_{\mathrm{d}}{w}_{\mathrm{d}}& =K_5-b_2\hfill \\ \hfill w_{\mathrm{d}}^2& =K_4-b_1-b_2·2{\xi }_{\mathrm{d}}{w}_{\mathrm{d}}\hfill \\ \hfill {\lambda }_2& =-(K_3-b_2·w_{\mathrm{d}}^2-b_1·2{\xi }_{\mathrm{d}}{w}_{\mathrm{d}}-b_0)/\mathsf{\Gamma }\hfill \\ \hfill k_{\mathrm{t}}& =K_0/(\mathsf{\Gamma }·w_{\mathrm{d}}^2)\hfill \\ \hfill {\lambda }_1& =-(K_2-b_{\mathrm{l}}·w_{\mathrm{d}}^2-b_0·2{\xi }_{\mathrm{d}}{w}_{\mathrm{d}}-k_{\mathrm{t}}·\mathsf{\Gamma })/\mathsf{\Gamma }\hfill \\ \hfill {\lambda }_0& =-(K_1-b_0·w_{\mathrm{d}}^2-k_{\mathrm{t}}·\mathsf{\Gamma }·2{\xi }_{\mathrm{d}}{w}_{\mathrm{d}})/\mathsf{\Gamma }.\hfill \end{array}$$

(22)

Thus, the parameters of the PAVPF damping controller and the tracking controller can be calculated simultaneously. The results are shown in

Table 3. Controller parameters.

	PEA1	PEA2	PEA3
${\lambda }_2$	−0.3407	−0.4274	−0.4758
${\lambda }_1$	874.5	590.8	560.8
${\lambda }_0$	1.569 × 106	1.35 × 106	1.414 × 106
$2{\xi }_{\mathrm{d}}{w}_{\mathrm{d}}$	2956	2775	2847
$w_{\mathrm{d}}^2$	3.587 × 106	3.246 × 106	3.447 × 106
$k_{\mathrm{t}}$	308	327	300

:

Among them, the parameters of the integral tracking controller are increased by about 10 times compared to before using PAVPF damping control. The frequency response characteristics of the stage after PAVPF damping control are shown in Figure 10. It can be seen that the first resonance peak and other resonance effects are effectively suppressed.

4.3. Step Signal Tracking Experiment

In order to verify the effectiveness of PAVPF damping controller for resonance suppression, this paper uses a PVPF damping controller for comparison. The Butterworth filter pattern is also used to simultaneously design the parameters of the damping controller $C_{\mathrm{dv}}\left(s\right)$ and the tracking controller $C_{\mathrm{tv}}\left(s\right)$. The controllers are expressed as follows:

$$C_{\mathrm{dv}}\left(s\right)={\displaystyle \frac{{\gamma }_{2}s+{\gamma }_1}{s^2+2{\xi }_{\mathrm{v}}{w}_{\mathrm{v}}s+w_{\mathrm{v}}^2}},$$

(23)

$$C_{\mathrm{tv}}\left(s\right)={\displaystyle \frac{k_{\mathrm{v}}}{s}}.$$

(24)

As mentioned above, the PVPF damping controller is mainly used for the second-order system, so it is necessary to simplify the dynamic model of PEAs to the second-order system $G_{\mathrm{s}}\left(s\right)$, expressed as

$$G_{\mathrm{s}}\left(s\right)={\displaystyle \frac{\delta }{s^2+2{\xi }_{\mathrm{m}}{w}_{\mathrm{m}}s+w_{\mathrm{m}}^2}}.$$

(25)

The simplified second-order system parameters are shown in

Table 4. Transfer function parameters.

	${\mathit{G}}_{\mathbf{s}1}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathbf{s}2}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathbf{s}3}\left(\mathit{s}\right)$
$\delta$	1.34 × 106	1.13 × 106	1.219 × 106
$2{\xi }_{\mathrm{m}}{w}_{\mathrm{m}}$	39.01	77.35	32.73
$w_{\mathrm{m}}^2$	1.67 × 106	1.478 × 106	1.493 × 106

. Then, the closed-loop transfer function of the system can be calculated by Equation (18), and the results are shown in Equations (25)–(27). By equating $D_{\mathrm{v}}\left(s\right)$ with the characteristic equation of the fifth-order Butterworth filter shown in Equation (28), the parameters of the controllers can be obtained. ${\xi }_4$ and ${\xi }_5$ are the desired closed-loop damping ratios, ${\xi }_4$ = $cos(\pm {36}^{\circ })$ = 0.8090, ${\xi }_5$ = $cos(\pm {72}^{\circ })$ = 0.3090, the poles are evenly distributed in the left half plane, and each pole of the ideal fifth-order system is ${36}^{\circ }$ apart. The ideal closed-loop system pole distribution is shown in Figure 9, from which the parameters of the controller can be obtained, and the results are shown in

Table 5. Controller parameters.

	PEA1	PEA2	PEA3
${\gamma }_2$	−2528	−2169	−2381
${\gamma }_1$	3.48 × 106	3.036 × 106	3.077 × 106
$2{\xi }_{\mathrm{v}}{w}_{\mathrm{v}}$	4104	3780	3889
$w_{\mathrm{v}}^2$	6.915 × 106	5.97 × 106	6.199 × 106
$k_{\mathrm{v}}$	389	393	361

.

$$G_{\mathrm{sys}}^{\mathrm{pvpf}}\left(s\right)={\displaystyle \frac{C_{\mathrm{dv}}\left(s\right)G_{\mathrm{s}}\left(s\right)}{1-G_{\mathrm{s}}\left(s\right)(C_{\mathrm{dv}}\left(s\right)-C_{\mathrm{tv}}\left(s\right))}}={\displaystyle \frac{N_{\mathrm{v}}\left(s\right)}{D_{\mathrm{v}}\left(s\right)}},$$

(26)

$$N_{\mathrm{v}}\left(s\right)=k_{\mathrm{tv}}\delta (s^2+2{\xi }_{\mathrm{m}}{w}_{\mathrm{m}}s+w_{\mathrm{m}}^2),$$

(27)

$$\begin{array}{cc}\hfill D_{\mathrm{v}}\left(s\right)& =s^5+(2{\xi }_{\mathrm{m}}{w}_{\mathrm{m}}+2{\xi }_{\mathrm{v}}{w}_{\mathrm{v}})s^4+(w_{\mathrm{m}}^2+w_{\mathrm{v}}^2+2{\xi }_{\mathrm{m}}{w}_{\mathrm{m}}·2{\xi }_{\mathrm{v}}{w}_{\mathrm{v}})s^3\hfill \\ \hfill & +(2{\xi }_{\mathrm{m}}{w}_{\mathrm{m}}·w_{\mathrm{v}}^2+2{\xi }_{\mathrm{v}}{w}_{\mathrm{v}}·w_{\mathrm{m}}^2-\delta {\gamma }_2+k_{\mathrm{tv}}\delta )s^2\hfill \\ \hfill & +(w_{\mathrm{m}}^2{w}_{\mathrm{v}}^2-\delta {\gamma }_{\mathrm{l}}+k_{\mathrm{tv}}\delta ·2{\xi }_{\mathrm{v}}{w}_{\mathrm{v}})s+k_{\mathrm{tv}}\delta w_{\mathrm{v}}^2,\hfill \end{array}$$

(28)

$$B\left(s\right)=(s+w_{\mathrm{b}})(s^2+2{\xi }_4{w}_{\mathrm{b}}s+w_{\mathrm{b}}^2)(s^2+2{\xi }_5{w}_{\mathrm{b}}s+w_{\mathrm{b}}^2).$$

(29)

The step signal can easily excite the resonance characteristics of the stage, so the step signal tracking experiment shown in Figure 3b was also conducted, and an amplitude of 1 V was applied to the three PEAs, respectively. The results are shown in Figure 11. It can be seen that not only the tracking speed is significantly improved, the stability time of PVPF and PAVPF is increased to 0.02 s, and the resonance is significantly suppressed. At the same time, compared with PVPF, the overshoot of PAVPF is reduced from 15% to 4%, which proves the effectiveness of PAVPF damping control and parameter design methods.

5. Experiment

5.1. Composite Control Method

In order to further improve the tracking accuracy and tracking speed, and reduce the influence of hysteresis and resonance characteristics to the stage, this paper adopts a composite control method of IMFC, PAVPF damping control and integral feedback tracking control to control the positioning stage. The composite control block diagram is shown in Figure 12. Firstly, the influence of hysteresis nonlinearity is eliminated by IMFC, and then the damping of the system is improved by PAVPF damping control, which also suppresses the influence of resonance. Finally, combined with the integral feedback tracking controller, the tracking speed of the stage is further improved to achieve high-precision positioning.

5.2. Triangle Wave Signal Trajectory Tracking Experiment

In order to verify the effectiveness of the above composite control method, a triangular wave signal trajectory tracking experiment is carried out on the stage, and the compensation effects of different control methods are compared. A triangular wave signal with an amplitude of 1V and a frequency of 5 Hz is input to each PEA. The trajectory tracking results and tracking errors are shown in Figure 13. At the same time, the RMSE is also used to judge the tracking effect, and the results are shown in

Table 6. RMSE of triangle wave signal trajectory tracking experiment (Unit: µm).

RMSE	PEA1	PEA2	PEA3
IMFC	0.8815	0.7493	0.7813
PAVPF	0.5711	0.5447	0.5872
IMFC + PAVPF	0.1620	0.1669	0.1798

. It can be seen that compared with the simple IMFC or PAVPF damping control, the RMSE of the composite control method is reduced by at least 77% and 69%, respectively, and the maximum error has been reduced by an average of 70% and 54%, respectively, which significantly improves the tracking accuracy of the stage, proving the effectiveness of the proposed composite control method. In addition, the maximum trajectory tracking error of the composite control method is only 0.4813 µm, which is only 2% of the maximum tracking stroke, also proving the superiority of the composite control method. In summary, the composite control method can be practically applied to similar piezoelectric-driven stages and achieve high-precision positioning control.

6. Conclusions

In order to solve the hysteresis and resonance problems of PEAs, this paper takes the three-degree-of-freedom piezoelectric-driven nanopositioning stage as the object, converts the three-degree-of-freedom motion into three sets of PEA outputs through the kinematic model and then performs single-axis control. Firstly, the nonlinearity of PEAs is compensated by the inverse rate-dependent PI model, and the effectiveness of the model is verified by open-loop and closed-loop experiments. Then, for the third-order dynamic model, the parameters of the PAVPF damping controller and feedback integral tracking controller are designed based on the Butterworth filter pattern, which suppresses the resonance of the system and improves the tracking accuracy of PEAs. The effectiveness of the designed controller parameters is verified by step signal tracking experiment. Then, the above control methods are combined to perform composite control on the stage and conduct experiments on tracking the trajectory of triangular wave signals. Compared with the simple IMFC or PAVPF damping control, the RMSE of this composite control method is reduced by at least 77% and 69%, and the maximum error is reduced by an average of 70% and 54%, which proves the effectiveness of the composite control method and the accuracy of parameter design. This parameter design method can also be applied to other third-order systems.