*4.1. Step Response*

The step response is generally utilized to test the system's positioning accuracy. Further, it can also show the system's tracking performance for non-continuous trajectories. For step response, the gains of the proposed single-neuron adaptive controller are tuned to be *K* = 0.0002 and *d* = 0.4 following the method provided in Section 3.2. The gains of the PID controller are manually tuned to be *Kp* = 0.8, *Ki* = 1000, and K*d* = 0 following the critical ratio method. Step response experiments are carried out and the experimental results are shown in Figure 4. The step response of the open-loop system is also provided for the purpose of comparison.

**Figure 4.** Step response performance.

As the resonant frequency of the PEA is 69 kHz and the sampling rate is set to 10 kHz, the transient state of the open-loop system is not observable. However, the slow creeping of the PEA can be observed and the steady-state positioning error is quite large. As a result, for the PEA, the response is fast, whereas the steady-state positioning accuracy is low. For the PID control, the PEA can converge within 8 ms and the steady-state positioning error is reduced to the noise level, whereas oscillations can be found in the transient state. For the proposed single-neuron adaptive controller, the rise time is on the same level of the open-loop system, indicating a fast response. There are no oscillations in the transient state. The proposed controller can also reduce the steady-state positioning error to the noise level. The convergence time, i.e., the learning time, of the proposed controller is on the level of several milliseconds. The experimental results show that the learning time of the proposed controller has an obvious sensitivity to the magnitude of the step size. For a smaller magnitude step, e.g., 4 μm step, the proposed controller can respond quickly, whereas it will take a relatively longer time (approximately 14 ms) for the controller to converge. For larger magnitudes, e.g., 8 μm step, the proposed controller can respond quickly and converge in about 6 ms.

Based on the above experimental results, both the proposed controller and the PID controller can reduce the steady-state positioning error to the noise level. The proposed controller is superior to the conventional PID controller in that it achieves a fast response and smooth transient state behavior. For the convergence time, the proposed controller converges faster for large step values, whereas the convergence time increases for small step values.

#### *4.2. Tracking of Sinusoidal Trajectories*

Sinusoidal trajectories are selected in this paper to verify the tracking performance of the proposed method on continuous trajectories. First, a low-frequency trajectory is adopted to tune the parameters of the controllers. In this case, the rate-dependence of the PEA's hysteresis can be neglected. This helps to ease the tuning of the parameters. The parameters of the controllers are tuned until excellent tracking performance is achieved in this case. These values are then fixed and higher-frequency trajectories are utilized to test the robustness and adaptability of the controllers.

In this paper, a 1 Hz sinusoidal trajectory is utilized to tune the parameters of the proposed method and the PID controller. For the proposed method, the parameters are tuned to *K* = 0.002 and *d* = 0.4. For the PID controller, the gains of *Kp*, *Ki*, and K*d* are tuned to 1.11, 100, and 0, respectively. Figure 5 shows the tracking performance of the 1 Hz sinusoidal trajectory. Compared to the open-loop system, both the proposed method and the PID controller can successfully compensate the hysteresis of the PEA. The PEA can follow the desired trajectory well. It can be observed that the steady-state tracking error of the proposed method can be reduced to the noise level. The steady-state tracking error of the PID controller is slightly higher than the noise level but is still comparable to the proposed method. In the following experiments, sinusoidal and triangular trajectories with higher frequencies are utilized while the parameters of the two controllers are fixed to the above values. This helps to test the robustness and adaptability of the proposed method against the rate-dependence of the PEA's hysteresis.

**Figure 5.** Tracking of the 1 Hz sinusoidal trajectory: (**a**) Time plot and (**b**) tracking error.

Figures 6 and 7 show the sinusoidal trajectory tracking results at 10 and 50 Hz, respectively. As the frequency of the desired trajectory increases, the trajectory tracking error of the proposed method increases slightly but still remains at the same magnitude of the measurement noise, exhibiting high robustness and adaptability against the rate-dependence of the PEA's hysteresis. On the contrary, the tracking performance of the PID controller starts to drop significantly at the frequency of 10 Hz. For the 50 Hz trajectory, the tracking performance of the PID control is even lower than the open-loop control. As a result, the PID gains tuned at the 1 Hz trajectory is only applicable for slow trajectories and might not work properly for fast trajectories.

**Figure 6.** Tracking of the 10 Hz sinusoidal trajectory: (**a**) Time plot and (**b**) tracking error.

**Figure 7.** Tracking of the 50 Hz sinusoidal trajectory: (**a**) Time plot and (**b**) tracking error.

More trajectory tracking experiments are performed, whereas not all the experimental results are presented for the conciseness of this paper. In order to quantitatively investigate the tracking performance, the root-mean-square tracking error (RMSE) and relative root-mean-square error (RRMSE) of these sinusoidal trajectories are calculated and presented in Table 1. RMSE and RRMSE are defined in the following equations:

$$RMSE = \sqrt{\sum\_{i=1}^{N} \left( y\_i - r\_i \right)^2 / N} \,\tag{8}$$

$$RRMSE = \sqrt{\sum\_{i=1}^{N} \left(y\_i - r\_i\right)^2 / \sum\_{i=1}^{N} r\_i} \times 100\% \tag{9}$$

where *yi* and *ri* represents the *i* th values of the actual and desired trajectories, respectively, and *N* is the length of sampling data.


**Table 1.** Tracking errors of proportional-integral-derivative (PID) and the proposed approach.

As the parameters of the controllers are tuned at the 1 Hz trajectory, the RMSEs of the proposed method at the PID control are 64.5 and 101.9 nm, corresponding to a 0.76% and 1.2% relative error, respectively. As the frequency increases, the tracking performances of both controllers start to degrade. For the proposed method, the RMSE increases to 170.2 nm, i.e., 2.02% relative error. On the contrary, the RMSE of the PID control increases to 1158.8 nm, an approximately 13.63% relative error. These experimental results demonstrate that the proposed method can successfully compensate the rate-dependent hysteresis of the PEA.

#### *4.3. Tracking of Triangular Trajectories*

Sinusoidal trajectories are smooth trajectories that do not contain high-frequency harmonic components. In applications, triangular trajectories are also widely utilized, e.g., the scanning of the sample in an atomic force microscope. Unlike sinusoidal trajectories, triangular trajectories contain high-frequency harmonic components, increasing the di fficulty in control. Therefore, the triangular trajectory is also selected in this paper to verify the performance of the proposed method for non-smooth trajectories. The parameters of the PID controller and the proposed method obtained in Section 4.2 are also inherited.

As shown in Figure 8, for the 1 Hz triangular trajectory, the tracking performance is similar to the 1 Hz sinusoidal trajectory. Both the PID control and the proposed method can reduce the tracking error to the noise level. This demonstrates the applicability of the two controllers for slow trajectories. Similarly, the tracking performance of both controllers decreases with the increment in the frequency of the triangular trajectory, which is obvious in Figures 9 and 10. Due to the influence of the high-frequency harmonic components, the tracking performance of the proposed method on the triangular trajectories is slightly lower than that of the sinusoidal trajectories. However, the tracking error still remains at very small ranges, showing strong robustness against the rate-dependence.

**Figure 8.** Tracking of the 1 Hz triangular trajectory: (**a**) Time plot and (**b**) tracking error.

**Figure 9.** Tracking of the 10 Hz triangular trajectory: (**a**) Time plot and (**b**) tracking error.

**Figure 10.** Tracking of the 50 Hz triangular trajectory: (**a**) Time plot and (**b**) tracking error.

#### *4.4. Hysteresis Compensation E*ffi*ciency*

The trajectory tracking results for sinusoidal trajectories are utilized to analyze the hysteresis compensation efficiency of the proposed method. The hysteresis loops in these experimental results are shown in Figure 11, where a 45◦ line is included to show the unitary mapping from the desired trajectory to the actual trajectory. For the 1 Hz sinusoidal trajectory, the resultant input–output relationship of both the PID controller and the proposed method coincide with the 45◦ line. The hysteresis loop is not observable, which means the hysteresis has been efficiently compensated. As the frequency increases, the input–output relationship of the proposed method stays close to the 45◦ line. On the contrary, for the PID controller, obvious hysteresis loops can still be observed in the input–output relationship for sinusoidal trajectories higher than 5 Hz. From Figure 11, we can conclude that the proposed

method can successfully compensate the rate-dependent hysteresis of the PEA when compared to the PID controller.

**Figure 11.** Hysteresis compensation efficiency for the sinusoidal trajectories.
