**1. Introduction**

As a sub-nanometer-resolution actuation device, piezoelectric actuators (PEAs) have been widely applied in various applications requiring nanometer-accurate motion [1–4]. However, the inherent hysteresis nonlinearity of the PEA greatly degrades its positioning accuracy, thus affecting its applicability and performance in precise operation tasks. The most significant characteristics of the PEA's hysteresis are the rate-dependence and asymmetry [5–7], i.e., the hysteresis loop becomes thicker with the increment in the input rate (or frequency) and the hysteresis loop is not symmetric about the loop center. These characteristics increase the complexity of the system and cause grea<sup>t</sup> difficulties in hysteresis modeling and compensation.

To address the above problems, lots of control methods have been proposed to characterize and compensate the hysteresis of the PEA. Physical models can be derived from physical measurement methods, such as magnetization, stress–strain, and energy principles [8,9]. However, the mathematical representations are often complex, making it difficult to obtain the inverse hysteresis model. In the meantime, a phenomenon-based model is also proposed, such as the Preisach model [10], Prandtl–Ishlinskii (PI) model [11,12], and Maxwell model [13]. As the inversion of the classical PI model is analytically available, it has been widely utilized in much research to describe the hysteresis characteristics of the PEA. After the inversion model is obtained, it can be utilized as a feedforward hysteresis compensator. This modeling and inversion approach is widely adopted, and many adaptive

methods can be integrated [14–17]. In order to avoid the inversion calculation, the direct inversion method (DIM) is also proposed to identify the inverse hysteresis model directly from the measurements in parameter identification [18–20].

For model-based hysteresis compensation, the performance of the controller is highly dependent on the modeling accuracy of the hysteresis model. However, the PEA's hysteresis is susceptible to many factors, such as the external load and the frequency of the control input. This makes the modeling and compensation of the PEA's hysteresis very case-sensitive. As a result, a high-precision hysteresis model is generally difficult to obtain. Therefore, many intelligent control algorithms have been proposed to achieve higher robustness and adaptability. For instance, sliding mode control has been proposed to improve the accuracy and the robustness against noise and disturbances [14,21]. A linearization control method with feedforward hysteresis compensation and proportional-integral-derivative (PID) feedback has also been proposed [22]. Besides, iterative learning control schemes have been verified to achieve high-performance tracking for PEAs [23].

In the field of intelligent control, the neural network is a highly powerful system identification tool. It has a strong self-learning ability and powerful mapping ability to nonlinear systems, which has been widely used in the control of complex systems [24,25]. In the hysteresis compensation of the PEA, Wang and Chen presented a novel Duhem model based on the neural network to describe the dynamic hysteresis of PEAs [26]. An inversion-free predictive controller was proposed based on a dynamic linearized multilayer feedforward neural network model [27]. A cerebellar model articulation controller neural network PID controller was also proposed [4]. A radial basis function (RBF) network was also used to model and compensate for the PEA's hysteresis [28]. However, the use of the S-type action function increases the calculation difficulty for fast, high-frequency, and fast-response systems such as the PEA.

Among the neural network-based controllers, the single adaptive neuron system retains the advantages of the neural network and can satisfy the requirements of the real-time control of fast processes [29,30]. Therefore, a single-neuron adaptive hysteresis compensation method is proposed in this paper. The controller imitates an adaptive single-neuron system to learn and uses Hebb learning rules and supervised learning to adjust the controller. The controller can respond quickly to time-varying signals, making it suitable for the rate-dependent hysteresis compensation. Positioning and trajectory tracking experiments are carried out to investigate the performance of the proposed method. The performance of the PID control is also investigated for the purpose of comparison. For the positioning control, the proposed method can converge in about 8 ms and the steady-state tracking error can be reduced to the noise level of the system. For trajectory tracking, sinusoidal and triangular trajectories with frequencies up to 50 Hz are utilized. The experimental results show that the proposed method has excellent robustness and adaptability against the rate-dependence of the PEA's hysteresis, and the hysteresis can be successfully compensated.

This paper is organized as follows: Section 2 introduces the properties of the inherent hysteresis of the PEA. Section 3 presents the single-neuron adaptive controller design and analysis. To investigate the efficiency of the proposed method, experimental verifications and performance analyses are provided in Section 4. Section 5 summarizes this paper.

#### **2. The Hysteretic Nonlinearity of the PEA**

## *2.1. Experimental Setup*

As shown in Figure 1, in this paper, a standalone PEA (model PZS001 from Thorlabs with integrated strain gauge sensors, Newton, NJ, USA) with a high-voltage amplifier (model ATA-4052 from Aigtek with a bandwidth of DC-500 kHz, Xi'an, China) is selected as the plant. The maximum displacement output of the PEA is measured to be 12.925 μm under the maximum actuation voltage of 10 V, i.e., the actuation gain (displacement/voltage) is 1.2925. According to the datasheet, the resonant frequency of the PEA used in this paper is 69 kHz. A dynamic Wheatstone bridge amplifier (model

SDY2105 from Beidaihe Institute of Practicality Electron Technology with a bandwidth of DC-300 kHz, Qinhuangdao, China) is utilized to measure the strain of the PEA, which is used to calculate the displacement of the PEA. The data acquisition and closed-loop control tasks are implemented on a real-time target (model microlabbox from dSPACE, Paderborn, Germany) with a sampling rate of 10 kHz. The algorithm is programmed in Simulink and implemented in Controldesk. Due to the influence of the strain gauges and the Wheatstone bridge amplifier, the measurement noise of the overall system is found to be ±34 nm.

**Figure 1.** Schematic of the system setup for the standalone piezoelectric actuator (PEA).

#### *2.2. Characteristics of the PEA's Hysteresis*

Obvious nonlinearities can be observed in the input–output relationship of the PEA. Generally, the hysteresis is the dominant factor affecting the motion accuracy of the PEA. This paper uses sinusoidal signals of *u*(*t*) = 5sin(2π*ft* − π/2) + 5 to drive the PEA at different frequencies. By observing the input signal and the measured displacement output of the PEA, hysteresis loops of the PEA can be obtained. Figure 2 depicts the measured input–output loops of the standalone PEA. As the resonant frequency of the PEA is 69 kHz, within the driving frequency of 1–400 Hz, the dynamics of the PEA can be neglected. As a result, the measured input–output loops shown in Figure 2 are totally produced by the hysteretic nonlinearity of the PEA. It can be seen that there is an obvious rate-dependent behavior in the measured hysteresis loops. As the input frequency increases, the hysteresis loop grows bigger and thicker. In addition, the hysteresis loop is not strictly symmetric about the loop center. The above rate-dependence and asymmetry properties increase the model complexity and increase the difficulty in the controller design of the PEA. Therefore, how to compensate the hysteresis and linearize the system have become crucial problems of the PEA.

**Figure 2.** The measured hysteresis loops of the standalone PEA.

#### **3. Single-Neuron Adaptive Controller Design**

#### *3.1. Single-Neuron Adaptive Control Algorithm*

Aiming to compensate the hysteresis of the PEA, this paper proposes a single-neuron adaptive controller without modelling the hysteresis of the PEA. A single neuron is a non-linear processing unit that has self-learning and self-adaptive capabilities and is applicable for many different control tasks. The input and output of a single-neuron system are expressed as follows:

$$y = K \cdot \sum\_{i=1}^{n} w\_i \mathbf{x}\_i + \delta\_\prime \tag{1}$$

where *K* denotes the gain characterizing the response speed of a neuron; *xi*, *y*, and δ are the state variable, output, and threshold, respectively; and *wi* represents the weight of *xi* that can be adjusted by the learning rules.

Neurons are generally considered to be self-organizing by modifying their synaptic weighting values. Supervised Hebb learning rules are usually used for the adjustment of the weights. Assuming the weight of the neuron *wi*(*t*) during learning is proportional to the signal *pi*(*t*) and decays slowly, the learning rule of the neuron can be expressed as

$$w\_i(t+1) = (1-c)w\_i(t) + dp\_i(t),\tag{2}$$

where *c* is a positive constant that determines the impact of the last weight value, *d* is a constant characterizing the learning efficiency, and *pi*(*t*) is the learning rules. To further improve the adaptability of neurons, the following learning rules are employed:

$$p\_i(t) = Z(t)S(t)\mathbf{x}\_i(t),\tag{3}$$

where *S*(*t*) indicates that the adaptive neuron adopts the Hebb learning rule, and *Z*(*t*) shows supervised learning rules. *Z*(*t*) means that external information is self-organized to have a control effect under the guidance of the teacher signal. In this way, the adaptive neuron algorithm combined with Hebb learning rules and supervised learning can perform self-organizing and adaptive control for nonlinear systems.

#### *3.2. Controller Design for the PEA*

As shown in Figure 3, the state variables to the controller are calculated by the error between the desired trajectory *r*(*t*) and the actual trajectory *y*(*t*). The output of the controller is *u*(*t*). In order to ensure the convergence and robustness of the learning algorithm, the following modified adaptive learning algorithm is adopted in this paper:

$$\begin{aligned} \mathbf{x}\_1(t) &= \mathbf{c}(t) \\ \mathbf{x}\_2(t) &= \Delta \mathbf{x}\_1(t) = \mathbf{c}(t) - \mathbf{c}(t-1) \\ \mathbf{x}\_3(t) &= \Delta \mathbf{x}\_2(t) = \mathbf{c}(t) - 2\mathbf{c}(t-1) + \mathbf{c}(t-2) \end{aligned} \tag{4}$$

where *e*(*t*) = *r*(*t*) − *y*(*t*) is the error between the desired and actual trajectories, and *x*1(*t*), *x*2(*t*), and *x*3(*t*) are adopted as the state variables to the neuron system.

**Figure 3.** Schematic diagram of the single-neuron adaptive hysteresis compensation method.

The previous controller output *<sup>u</sup>*(*<sup>t</sup>*−1) can be utilized as the threshold, i.e., δ = *u*(*<sup>t</sup>* − 1). Substituting this into Equation (1), the controller output of the single-neuron adaptive controller can be written as follows:

$$u(t) = K \cdot \sum\_{i=1}^{n} w\_i x\_i + u(t-1),\tag{5}$$

For the adaption of the weights, *Z*(*t*) = *e*(*t*) is adopted as the supervisory function and *S*(*t*) = *u*(*t*) is adopted as the Hebb learning rule. Substituting these into Equations (2) and (3), the learning rule of the neuron can be expressed as follows:

$$w\_i(t) = w\_i(t-1) + d \cdot e(t) \cdot u(t) \cdot x\_i(t),\tag{6}$$

where *c* is set as 0 because *wi*(*t*) will converge to a stable value if *c* is small enough. According to the common experience of single-neuron adaptive control, *d* is typically less than 0.5. In this paper, *d* = 0.4 is adopted.

The whole control progress proceeds as follows. After getting the desired trajectory and actual trajectory, the state variable *x*i(*t*) is calculated using Equation (4). Three state variables correspond to three control outputs produced by the neuron, which are the proportional feedback *u*1(*t*), first-order differential feedback *u*2(*t*), and second-order differential feedback *u*3(*t*), respectively. The proportional feedback can quickly reduce the tracking error. The first-order differential feedback can improve the system's transient state performance, i.e., the response speed and overshoot. The second-order differential feedback ensures that the system remains stable during a fast response. The change in the weight reflects the dynamic characteristics of the controlled plant and the response process. The neuron continuously adjusts the weight through its own learning rules, and quickly eliminates the error and enters the steady-state under the correlation of the three feedbacks.

The system response speed is positively proportional to *K*, but a large overshoot might make the system unstable. On the contrary, if *K* is too small, the actual trajectory cannot track the desired trajectory. Thus, the tuning of *K* is very important. In order to determine the proper value for *K*, we built a mathematical model for the PEA using the Prandtl–Ishlinskii model. Through several simulation tests, the influence of *K* is computationally investigated. A candidate *K* is then selected according to the simulation results. Subsequently, this candidate *K* is adopted as the initial value and it is tuned manually online to achieve improved tracking performance. In this case, only fine tuning within a very small range is necessary.

#### **4. Experimental Verifications**

On the basis of the above analyses, the single-neuron adaptive control algorithm is applied to compensate the PEA's hysteresis. Positioning and trajectory tracking experiments are carried out to verify the proposed method's performance in hysteresis compensation.

In order to better compare the performances, this paper also includes the open loop and PID control results for the purpose of comparison. For the open-loop control, the PEA is assumed to be linear and the actuation gain (the ratio between the maximum allowable control input and the maximum displacement output) is utilized to finish the input–output mapping, i.e.,

$$u(t) = r(t) \cdot \frac{\mathcal{U}\_{\text{max}}}{\mathcal{Y}\_{\text{max}}} \, \, \, \tag{7}$$

where *U*max and *Y*max are the maximum allowable control input and maximum displacement output, respectively. The open-loop control represents the basic characteristics of the system as no controller is utilized.

PID control, a widely utilized controller, has the advantages of simple parameter adjustment and ease of use. However, for nonlinear systems such as the PEA, the tuning of the gains in the PID controller is not an easy task. It might not work properly to systematically adjust the PID gains via strictly following well-developed approaches such as the Ziegler–Nichols method. Further, the behavior of the PEA is susceptible to many factors, making it di fficult or impossible for the PID control to maintain the control performance in all scenarios. All these increase the di fficulty in PID tuning. In this paper, the critical ratio method is adopted to tune the PID gains. At the beginning, only the proportional gain *Kp* is tuned with the other gains set to 0. Subsequently, the other gains are adjusted after the *Kp* is specified. For the PEA, PI control is found to be adequate to achieve satisfactory performance. In fact, a trial and error process is inevitable to finely tune the PID gains to achieve satisfactory results.
