Modeling and Tracking Control for Piezoelectric Actuator Based on a New Asymmetric Hysteresis Model

Ⅰ. INTRODUCTION

PIEZOELECTRIC actuators are often used in precision positioning and tracking control application, such as scanning and microscopic technology [1], [2], piezo-actuated flexure mechanisms [3], [4], deformable mirrors [5], optical deflectors [6], fast steering mirrors [7], and MEMS (micro-electromechanical systems) microscanners [8]. This is because of their nanometer displacement resolution, fast frequency response, and high stiffness. However, one of the main obstacles degrading the performance of actuator in positioning control applications is the inherent hysteresis nonlinearity of the displacement to the applied voltage. Hysteresis in piezoelectric material is a nonlocal memory effect. This means that the future values of the hysteresis output depend not only on the current values of input and output, but also on the past extremal values of the input signal.

Due to the non-smooth, non-differentiable and non-memoryless characteristics, the hysteresis nonlinearity could give rise to undesirable inaccuracy in open-loop positioning system or inadvertent oscillations of the output [9], [10] in closed-loop system without the aid of further control. Therefore, tremendous amount of control methods have been proposed to eliminate or compensate for the hysteresis effect. They can be broadly classified into three categories: 1) electric charge control [11]; 2) closed-loop displacement control; 3) model-based feed-forward control. For the first category, hardware realization is very complicated, and thus there has been little effective discussion about this method. Most commercial systems (e.g., Physik Instrumente, Inc.) fall into the second category and can achieve nano-scale positioning precision. This category consists of many different approaches, such as sliding model control [12], adaptive control [13], neural network control [14], proportional-integral control with inverse compensation [15], $H_\infty$ control [16] and so on. The key ideas of feed-forward methods include: 1) building a hysteresis model which is very similar to the real hysteresis curve, and 2) realizing a feed-forward controller based on an inverse model to linearize the response of the actuators. However, the challenge of the model-based feed-forward control is model accuracy and computational complexity [17].

In order to eliminate hysteresis using appropriate control method, the description or modeling for hysteresis is inevitable. Many hysteresis models have been reported, such as Maxwell model [18], support vector machine model [19], Duhem model [20], Bouc-Wen model [21], Prandtl-Ishlinskii (PI) model [22] and so on. Among these hysteresis models, the PI model is very suitable for real time application because it has simpler implementation procedures and has analytical inverse [23]. Using PI approach can reduce maximum hysteretic error to about 1%$-$3% in open-loop control with quasi-static tracking [24] and to about 1% in tracking of a non-stationary constant-rate saw-tooth profile with closed-loop adaptive control [25].

However, on the basis of laboratory measurements, it has been shown that hysteresis curve in our lead zirconium titanate (PZT) actuator is highly asymmetric and concave about the origin, as in Fig. 1. Apparently, the conventional (CPI) model is not well suitable for our PZT Prandtl-Ishlinskii actuator. To overcome the drawback of the CPI model, researchers have proposed a variation [26] which can describe asymmetric hysteresis loops by adding a one-side deadzone operator, but this model is mathematically complicated and is difficult to calculate and implement in practice. And there are some other papers [27], [28] which also describe the asymmetric properties. But [27] could not realize hysteresis curves to be concave at the origin which is revealed in our experimental data. In [28] the hysteresis model is constructed and validated based on shape memory alloy (SMA) actuator other than piezoelectric actuator, and the model output strongly depends upon appropriate envelope function which is not easy to find in practice. Here, a novel asymmetric PI model is proposed in this paper to characterize the asymmetric hysteresis and is experimentally validated to be effective in practice.

Figure  1.  Illustration hysteresis curve obtained from experiments.

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The goal of the present paper is to experimentally demonstrate the applicability of this new modeling and tracking control method to a special class of hysteretic nonlinearities system especially to piezoelectric actuator for low-frequency real-time trajectory tracking application. Moreover, this method could be extended to the underlying systems which are involved with any switching dynamics [29]$-$[31] in the future.

This paper is organized as follows. Section Ⅰ presents the literature review and research motivation for the control of PZT actuators. The conventional play-operator-based PI model is described in Section Ⅱ. The proposed modified asymmetric PI model is detailed in Section Ⅲ. Section Ⅳ compares the modeling accuracy of the modified PI model with the conventional one. Section Ⅴ presents the performance evaluation experiments for the application of the new modeling and tracking control method to piezoelectric actuator. In Section Ⅵ, conclusions are drawn.

Ⅱ. THE CPI MODEL

One of the advantages of PI type hysteresis model is that it is purely phenomenological; there are no direct relationships between modeling parameters and the physics nature of the hysteresis. Therefore, we would model the hysteresis with reference only to the experimental observations.

This section describes the Prandtl-Ishlinskii modeling method of hysteresis proposed by Kuhnen et al. [24], where a play hysteresis operator is used. Before presenting the conventional play-operator-based Prandtl-Ishlinskii model, we shall introduce the essential well known play operator in order to derive formulations for the PI hysteresis model. The play operator is a continuous rate-independent operator. It is called a play operator because it was used to describe the mechanical play at the beginning. A detailed discussion about this operator can be found in [32]. The input-output behavior of a play operator with a threshold value $r > 0$ is given by the hysteresis diagram depicted in Fig. 2.

Figure  2.  Input-output relationship of play operator.

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In the discrete-time domain, a play operator $H_r$ is defined by

where $x$ is the control input, $y$ is the output response, $r$ is the threshold value of control input, and $T$ is the sampling period, $y_0$ is the initial value of the output, $t$ is the time. Alternatively, (1) can be written for ascending, constant, and descending output, respectively, as follows:

The initial consistency condition of (2) is given by

with $y_0\in {\mathbb R}$, and is usually but not necessarily initialized to zero.

Then, complex hysteretic nonlinearity $y(t)$ can be modeled by a linearly weighted superposition of many play operators with different threshold and weight values

where weight vector is ${ w}^{T}$ and play operators are ${ H}_r[x, y_0](t)$ with the threshold vector ${ r}=[r_1, \ldots, r_n]^{T}$ where $0=r_1<$ $\cdots$ $<$ $r_n$, and the initial state vector ${y}_0=[y_{\tiny 01}, \ldots, y_{\tiny 0n}]^{T}$. The control input threshold values ${r}=[r_1, \ldots, r_n]^{T}$ are distributed within the range not exceeding the maximum amplitude of the input signal and are usually but not necessarily chosen to be equal intervals. The CPI model is shown in Fig. 3. Apparently, the curve is symmetric and convex.

Figure  3.  CPI model.

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Ⅲ. THE MPI MODEL

Modeling of piezoelectric actuator with the CPI hysteresis operator lacks accuracy due to the rigid structure of the primary play operator. For example, the PI model consists of many play operators which are symmetric about a center point, and hence inherits symmetry properties. But, in reality, most hysteresis loops are asymmetric. Using a symmetric model to describe an asymmetric hysteresis loop will lead to inherent systematic errors. The other shortfall of this operator is its lack of accuracy in adjusting the residual displacement around the origin as shown in Fig. 1. The motivation of this work is to find a new approach that can precisely model the asymmetric hysteresis of piezoelectric actuators. Here, the proposed asymmetric PI model is discussed below.

A Modeling Method of MPI

A representative hysteresis curve of our PZT actuator is shown in Fig. 1. It is highly asymmetric and concave at the origin, and the CPI model cannot exactly describe this character. Inspired by experimental data, a novel modified PI modeling method is proposed to solve the problem in this paper. In the method, a coupled-play (CP) operator shown in Fig. 4 is used as the elementary operator. The CP operator which is combination of two half-play operators can be formulated as:

Figure  4.  Input-output relationship of CP operator.

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where $ra$, $rb$, $rc$ and $r$ are the thresholds of the CP operator; $x(k)$ represents the current input; $x_0(k-1)$ and $z_0(k-1)$ represent previous input and output, respectively.

Generally, the initial consistency condition of (5) is: $x_0(k$ $-$ $1)$ $=$ $0$; $z_0(k-1) = 0$. Hence, the corresponding modified PI model can be described by

where ${{w}}={[{w_1}, \ldots, {w_n}]^{T}}$ is the weight vector; ${ {H}}[x(k),$ ${{{ x}}_{{0}}}(k-1), {{ {z}}_{{0}}}{ {(k-1)}}, { {r, ra, rb, rc}}](k)$ is the vector of the CP operators; ${{{x}}_{{0}}} = {[{x_0}_1, \ldots, {x_0}_n]^{T}}$ and ${{{ z}}_{{0}}} =[{z_0}_1, \ldots, {z_0}_n]^{T}$ are the initial state vectors; ${{r}} = {[{r_1}, \ldots, {r_n}]^{T}}$, ${{ra}} = [r{a_1}, \ldots,$ $r{a_n}]^{T}$, ${ {rb}} = {[r{b_1}, \ldots, r{b_n}]^{T}}$ and ${ {rc}} = {[r{c_1}, \ldots, r{c_n}]^{T}}$ are the threshold vectors.

In this modified hysteresis model, a CP operator is used to replace the play operator. Different threshold vector can derive adjustable and asymmetric hysteresis curves.

B Illustration Example for Property Evaluation of CP Operator

In the proposed modified hysteresis model, a CP operator is used to replace the play operator. In this section, an example is used to evaluate the theoretical property of CP operator. The input-output relationship of the CP operator is supported by the numeric simulation shown in Fig. 5, which shows an input signal and the input-output trajectory of the CP operator corresponding to the input. In the beginning, the input is at point $A$ and quickly increases to its first maximum point $B$. The input-output trajectory follows the polygonal arrow line $A$ to $B$ in Fig. 5(b). After that, the input decreases to its first minimum point $C$, and this process corresponds to the polygonal arrow line $B$ to $C$ in Fig. 5(b). Next, the input increases to point $D$, and this process follows the polygonal arrow line $C$ to $D$ in Fig. 5(b). Then, the input decreases to point $E$, and the output follows the straight arrow line from $D$ to $E$. After that, the input increases to point $F$, and input-output trajectory follows the polygonal arrow line $E$, $D$ to $F$ in Fig. 5(b). After reaching F, the input decreases again and gets to point $G$, while this process takes the input-output trajectory of polygonal arrow line $F$ to $G$ in Fig. 5(b). Next, the input increases to point $H$, and the output follows the straight arrow line $G$ to $H$.

Figure  5.  Illustration of the output-input property for CP operator.

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From above process, we can see that the output path will be automatically selected according to the input process. It can be observed from the Fig. 5 that the output will hold to zero for a short time when the output arrives at zero. This phenomenon is different from the play operator, and it will make the output of MPI model to be concave about the origin. In addition, it will make the output of MPI model to be asymmetric when $ra$ $\neq$ $-r$ or $rb\neq-rc$. Therefore, the input-output relationship of the CP operator indeed can help make the MPI model possesses asymmetric property and concave property about the origin.

C Inverse of the MPI Hysteresis Model

The key idea of an inverse feed-forward controller is to cascade the inverse hysteresis model with the actual hysteresis to get an identity mapping between the desired actuator output and the real actuator response.

One of the advantages of the PI type hysteresis model is that its inverse is also of PI type, with different thresholds and weight values. Analogously, the inverse of the modified PI hysteresis model can be obtained as:

Ⅳ. MODEL VALIDATION AND EXPERIMENT RESULTS

A Experimental Setup

In order to validate the effectiveness of the proposed modified PI modeling method, an experimental platform is established as shown in Fig. 6 and the experimental architecture is shown in Fig. 7. In the experiment, the displacement output of piezoelectric actuator is measured with accuracy of 80 nm by the strain gages glued on its mechanical framework. The output signal of strain bridge was acquired, amplified and filtered by a dynamic bridge input module (PXIe-4331, NI Corp.) with 24-bit resolution A/D converters. The PXIe-8115 Platform (NI Corp.) equipped with LabVIEW software was used to implement the real-time control procedure for the piezoelectric actuator. And the control signal was outputted by a NI X Series Multifunction DAQ device (PXIe-6363) with 16-bit resolution D/A converters, and then amplified by a piezoelectric voltage amplifier (from IOE, CAS) before applying to piezoelectric actuator.

Figure  6.  Experiment setup.

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Figure  7.  Experimental architecture.

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B Identification of the Hysteresis Model

In this subsection, the proposed novel modified hysteresis model for piezoelectric actuator is investigated by a set of experimental tests. For a set of CP operators with predefined threshold values $r_i$, it is desired to identify the weighting parameters and appropriate values of $ra$, $rb$, $rc$, to obtain a minimal error between the experimental results and model response. For this purpose, 20 CP operators are exploited here to cover the input voltage range of $-500$ V to 500 V. Threshold values $r_i$ are chosen in an orderly increasing sequence with fine intervals.

A recursive least-square optimization technique which is fast and simple is utilized here for the error minimization. The identification input is designed such that it can cover the entire span of the actuator input. Fig. 8 and Table Ⅰ demonstrate the identification results and estimated parameter values. The maximum and mean-square error values are obtained as 1.02% and 0.002 $\mu$m, respectively.

Figure  8.  Identification of the MPI hysteresis model.

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Table  Ⅰ.  PARAMETER VALUES OF IDENTIFIED DIRECT MODEL USED IN (6)

$i$	$w$	$r$	$ra$	$rb$	$rc$
0	0.0043	0.000	0.000	0.000	0.000
1	0.0003	25.000	23.750	7.500	8.750
2	0.0003	50.000	47.500	15.000	17.500
3	0.0002	75.000	71.250	22.500	26.250
4	0.0002	100.000	95.000	30.000	35.000
5	0.0002	125.000	118.750	37.500	43.750
6	0.0002	150.000	142.500	45.000	52.500
7	0.0001	175.000	166.250	52.500	61.250
8	0.0001	200.000	190.000	60.000	70.000
9	0.0001	225.000	213.750	67.500	78.750
10	0.0001	250.000	237.500	75.000	87.500
11	0.0001	275.000	261.250	82.500	96.250
12	0.0001	300.000	285.000	90.000	105.000
13	0.0001	325.000	308.750	97.500	113.750
14	0.0002	350.000	332.500	105.000	122.500
15	0.0002	375.000	356.250	112.500	131.250
16	0.0002	400.000	380.000	120.000	140.000
17	0.0002	425.000	403.750	127.500	148.750
18	0.0003	450.000	427.500	135.000	157.500
19	0.0003	475.000	451.250	142.500	166.250

 | Show Table

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In order to demonstrate the effectiveness of the proposed modified PI model over the CPI model, a representative model of the CPI approach is developed. Fig. 9 depicts the hysteresis response of the CPI model to the same input shown in Fig. 8(a). Fig. 10 depicts the comparison of the modeling errors between the conventional and modified PI approaches. It is clearly observed that the modified PI model demonstrates better response than the CPI approach. The maximum and mean-square errors for the conventional model are obtained as 5.31% and 0.014 $\mu$m, respectively.

Figure  9.  Response of identified CPI model.

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Figure  10.  Modeling errors comparison of the CPI and MPI models.

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The thresholds and weighting values of the inverse hysteresis model are identified using (7), and the describing parameters of the inverse model are listed in Table Ⅱ.

Table  Ⅱ.  PARAMETER VALUES OF IDENTIFIED INVERSE MODEL USED IN (6)

$i$	$w'$	$r'$	$ra'$	$rb'$	$rc'$
0	234.9005	0.0000	0.0000	0.0000	0.0000
1	$-17.0350$	0.1064	0.0958	0.0372	0.0372
2	$-12.8939$	0.2212	0.1991	0.0774	0.0774
3	$-9.9779$	0.3431	0.3088	0.1201	0.1201
4	$-7.8667$	0.4714	0.4242	0.1650	0.1650
5	$-6.3131$	0.6050	0.5445	0.2117	0.2117
6	$-5.1645$	0.7432	0.6689	0.2601	0.2601
7	$-4.3238$	0.8855	0.7970	0.3099	0.3099
8	$-3.7267$	1.0315	0.9283	0.3610	0.3610
9	$-3.3295$	1.1806	1.0626	0.4132	0.4132
10	$-3.1008$	1.3328	1.1995	0.4665	0.4665
11	$-3.0166$	1.4879	1.3391	0.5208	0.5208
12	$-3.0567$	1.6460	1.4814	0.5761	0.5761
13	$-3.2021$	1.8072	1.6265	0.6325	0.6325
14	$-3.4337$	1.9718	1.7746	0.6901	0.6901
15	$-3.7312$	2.1402	1.9262	0.7491	0.7491
16	$-4.0730$	2.3129	2.0816	0.8095	0.8095
17	$-4.4364$	2.4907	2.2416	0.8717	0.8717
18	$-4.7987$	2.6742	2.4068	0.9360	0.9360
19	$-5.1381$	2.8644	2.5780	1.0025	1.0025

 | Show Table

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Ⅴ. PERFORMANCE EVALUATION OF TRACKING CONTROL WITH THE ASYMMETRIC PI MODEL

A The Scheme of the Hybrid Controller

Although the inverse model can be used as a feed-forward controller to perform an effective trajectory tracking, modeling errors and parameter uncertainties could degrade the accuracy of the model. Particularly in ultra-precision applications, the use of a feedback controller is inevitable. In order to achieve more efficient control with less tracking inaccuracy, a hybrid control strategy combining the feed-forward and feedback controller is designed to realize precise positioning. Fig. 11 presents the block diagram of the hybrid control strategy, where an inverse-model-based feed-forward controller connects in parallel a proportional-integral feed-back controller. The control algorithm is expressed as follows:

Figure  11.  The block diagram of the hybrid control scheme.

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where, $u(n)$ denotes the plant input, $K$ is the gain of the high voltage amplifier, $H^{-1}(n)$ represents the inverse hysteresis model, $e(n)$ is the error between a desired value $r(n)$ and the actual output $y(n)$, given as: $e(n) = r(n) - y(n)$.

Remark 1: It is important to note that over a broad range of frequencies, piezoelectric actuator is described as the cascade connection of the static hysteresis and second-order linear dynamics in many investigations, but in low-frequency operations, the effects of actuator damping and inertia can be safely neglected. In order to verify the effectiveness of proposed asymmetric hysteresis model and reduce the influence of dynamic effect, low-frequency operation range is selected where the dynamic effect does not work. Therefore, in this paper the motion equation of piezoelectric actuator is only treated as a static hysteresis relation between the input voltage and actuator displacement.

B Tracking Performance Comparisons of CPI and MPI Models

The experiment setup of the tracking control is identical to that in the parameter identification experiments. The control algorithm was implemented with a 10 kHz sampling rate. Under open loop operation, the input voltage fails to precisely control the output displacement of the piezoelectric actuator due to the hysteresis nonlinearity, and therefore the use of a feedback controller is inevitable. In order to suppress the tracking error between the real output and the desired output, two different hybrid controllers were utilized and compared in this section: 1) the first hybrid controller which combines a symmetric-PI-model-based feed-forward controller and a feed-back controller; 2) the second hybrid controller which combines an asymmetric-PI-model-based feed-forward controller and a feed-back controller.

In the tracking experiment for the first hybrid controller, Fig. 12 shows that a 1 Hz triangular trajectory wave with a 3.3 $\mu$m amplitude is provided, and the tracking error is bounded within $-0.10$ to $+0.11\, \mu$m. The hysteresis has been reduced to 1.8%. For the second hybrid controller, Fig. 13 shows that when a 1 Hz triangular trajectory input is provided, the tracking error is bounded within $-0.06$ to $+0.07\, \mu$m. The hysteresis has been reduced to 1.0% instead of 1.8%. From what are shown above, it is known that the second hybrid control strategy has revealed a relatively smaller error and derived a better tracking performance compared with the first one.

Figure  12.  Tracking performance of the first hybrid controller.

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Figure  13.  Tracking performance of the second hybrid controller.

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C Tracking Performance of Controller With the Asymmetric PI Model

In order to further evaluate the performance of the second hybrid controller, in this section the experiments for tracking control of the sine signal of different frequencies are implemented. As shown in Figs. 14-17, when the tracking experiment uses sine waves with increased frequencies (5 Hz, 10 Hz, 15 Hz, 20 Hz), the maximum tracking errors increase gradually ($0.098^\circ$, $0.159^\circ$, $0.224^\circ$, $0.293^\circ$). And the average values of hysteresis can reach 0.9%, 1.0%, 1.2%, 1.3%, respectively. We can see that the hysteresis can be tuned with the proposed model and hybrid controller. At the same time, with the increase of frequency, the error will increase slightly. It is mainly due to dynamic effect of mechanical system. So, these results suggest that the proposed methodology is effective to control the piezoelectric actuator with higher precision.

Figure  14.  Tracking performance of sine waves of 5 Hz.

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Figure  15.  Tracking performance of sine waves of 10 Hz.

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Figure  16.  Tracking performance of sine waves of 15 Hz.

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Figure  17.  Tracking performance of sine waves of 20 Hz.

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Ⅵ. CONCLUSION

The main contribution of this paper is to apply a novel asymmetric Prandtl-Ishlinskii model to describe the hysteresis of the actuator for low-frequency real-time trajectory tracking application.

This novel MPI model can characterize the asymmetric hysteresis loops and exhibit a very high accuracy in adjusting the residual displacement around the origin. The validity of the proposed model is demonstrated by comparing simulation results with experimental measurements.

In order to further evaluate the performance of the proposed model in closed-loop tracking application, two different hybrid control methods which experimentally demonstrate their performance under the same operating conditions, are compared to validate that the hybrid control strategy with proposed hysteresis model can mitigate the hysteresis nonlinearity more effectively and achieve better tracking precision. Moreover, signals of different frequencies are utilized to validate the performance of the hybrid control method with proposed hysteresis model.

From those results, it can be concluded that the proposed modeling and tracking control strategy can achieve efficient control of piezoelectric actuator and a desired tracking precision.