# Hybrid Adaptive Controller Design with Hysteresis Compensator for a Piezo-Actuated Stage
## 4 Results
### Tracking Performance under Different Reference Trajectories  
Then, the tracking performance was evaluated with different reference trajectories. Sinusoidal and triangular reference signals with different frequencies (f = 1, 10, 20, and 50 Hz) were adopted. All the signals had amplitudes of 36 \(\upmu\)m. The results are shown in Figures 6 and 7, and the results of quantitative analysis are presented in Table 2. The tracking error was evaluated according to the MAXAE and MAE via (31) and (32), respectively. \(Z_{d}(k)=y_{r}(k)\) and \(Z(k)=y(k)\) represent the actual and desired outputs of the P-AS, respectively. All the MAXAEs and MAEs for the proposed control method were smaller than those for the compensator based on the inverse BW model and MRAC under sinusoidal and triangular reference signals with frequencies of 1, 10, 20, and 50 Hz. In particular, the MAE of the raised controller was 66.5% and 12.5% smaller than those of the compensator and MRAC, respectively, for the 50 Hz sinusoidal reference trajectory. Figure 6: Tracking performance of the HAC method in comparison with the compensator and MRAC under sinusoidal reference trajectories of different frequencies: (**a**) tracking result at frequency of 1 Hz; (**b**) tracking result at frequency of 10 Hz; (**c**) tracking result at frequency of 20 Hz; (**d**) tracking result at frequency of 50 Hz. Compared with the adopted sliding mode tracking control adopted in [36], the HAC method improves the control effect significantly. When the frequency of the triangular \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Reference** & **Frequency** & **MAXAE/MAE of the** & **MAXAE/MAE of** & **MAXAE/MAE of the** \\
**Trajectories** & **(Hz)** & **Compensator (\(\upmu\))** & **the MRAC (\(\upmu\))** & **Proposed Method (\(\upmu\))** \\ \hline Sinusoidal & 1 & 0.9522/0.4930 & 0.2976/0.0523 & 0.1748/0.0357 \\ signal & 10 & 0.8934/0.4968 & 0.3078/0.1400 & 0.1509/0.0803 \\  & 20 & 1.1772/0.5604 & 0.3949/0.1336 & 0.2607/0.1318 \\  & 50 & 1.6635/0.8371 & 0.8603/0.2919 & 0.4150/0.2122 \\ \hline Triangular & 1 & 0.7741/0.2678 & 0.1603/0.0141 & 0.0659/0.0088 \\ signal & 10 & 1.1250/0.5689 & 0.3862/0.1129 & 0.1707/0.0939 \\  & 20 & 1.3742/0.7145 & 0.6140/0.2792 & 0.3488/0.2250 \\  & 50 & 1.9488/0.9441 & 0.9414/0.3617 & 0.4296/0.3165 \\ \hline \end{tabular}
\end{table} 
Table 2: Tracking performance of different controllers under different reference trajectories. Figure 7: Tracking performance of the HAC approach in comparison with the compensator and MRAC under triangular reference trajectories of different frequencies: (**a**) tracking result at frequency of 1 Hz; (**b**) tracking result at frequency of 10 Hz; (**c**) tracking result at frequency of 20 Hz; (**d**) tracking result at frequency of 50 Hz. reference signals was 1 and 10 Hz, the proposed control method attenuated the MAE from 1.18% and 1.96% to 0.02% and 0.26%, respectively. In addition, the MAE of the HAC method was smaller than that of the neural-network self-tuning control raised in [50]. When the frequency of the sinusoidal reference signals was 1, 10, and 20 Hz, the proposed control method reduced the MAE from 1.02%, 1.26%, and 1.85% to 0.49%, 0.42%, and 0.72%, respectively. Figure 8 shows a further evaluation of the tracking performance with different complex reference trajectories. Figure 8a presents a comparison of the tracking results with the desired mixed sinusoidal signal. As shown, the tracking errors of the raised control approach were significantly smaller than those of the compensator and MRAC. Figure 8b presents a comparison of the tracking results with the desired mixed triangular signal. As shown, the MAE of the proposed control method was 0.0095 \(\upmu\)m, which was 97.6% and 74.1% smaller than those of the compensator and MRAC, respectively. The experimental results also show that the proposed control method has better control performance for both complex harmonic wave signal and triangular signal.