# LADRC-Based Magnetic Field Measurement Method for a Nuclear Magnetic Resonance Rotation Sensor
## 5 Experimental Evaluation
### Parameter Tuning of LADRC-Based Methods  
In this experiment, the parameter tuning of the proposed LADRC-based method was considered. Considering the system in Equation (13), the system parameter \(\Gamma\) can be identified by a free inducing decay experiment. According to corollary 1, the parameter \(b\) can be set as \(b=\frac{K_{c}}{k_{c}-\Gamma}>0\). Thus, only two parameters \(K_{c}\) and \(\omega_{o}\) need to be tuned. The influence of the two parameters were tested here. In Figure 6, the parameter \(K_{c}\) was fixed to 3 mHz/deg (\(\omega_{c}=1.08\) Hz), and the parameter \(\omega_{o}\) was set from 1-3 \(\omega_{c}\). A step input signal (5 nT) was added to the magnetic field of the z-axis, and the responses of both loops were recorded. Figure 6a,b show that the bandwidth of the LESO directly affects the phase-tracking performance, Figure 6c,d show its influence on frequency-observing, and Table 2 is the settling times of the experiments. The settling time of both loops is obviously reduced with a wider bandwidth. However, the bandwidth of LESO cannot be infinite due to the digital sampling and delay. According to the tests carried out in this paper, the bandwidth could be set to \(3\omega_{c}\) (a faster feedback loop may be unstable) to obtain an acceptable feedback system. Figure 5: Experimental performance of the PLL-based and LADRC-based methods with a step input signal. (**a–c**) are the phase-loop step response comparisons, (**d–f**) are the frequency-loop step response comparisons and (**g–i**) are the observed magnetic field. In Figure 7, the parameter \(\omega_{o}\) was fixed to 2.16 Hz, and the parameter \(K_{c}\) was set from 1-5 mHz/deg. A step input signal (5 nT) was added to the magnetic field of the z-axis, and the responses of both loops were recorded. Figure 7a,b are the phase-tracking response signals, Figure 7c,d are the frequency-observing response signals of both isotopes, and Table 3 is the settling times of the experiments. As is shown in the test results, \(K_{c}\) played a dominant role in the phase-loop response, but not in the frequency-loop. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Settling Time (s)**} & \multicolumn{2}{c}{**Settling Time (s)**} \\  & \multicolumn{2}{c}{**Frequency Loop**} & \multicolumn{2}{c}{**Phase Loop**} \\ \cline{2-5}  & **129 Xe** & **131 Xe** & **129 Xe** & **131 Xe** \\ \hline \(\omega_{o}=\omega_{c}\) & 15.251 & 15.303 & 17.46 & 19.21 \\ \(\omega_{o}=1.5\,\omega_{c}\) & 13.325 & 13.319 & 15.6 & 16.07 \\ \(\omega_{o}=2\,\omega_{c}\) & 12.337 & 12.271 & 14.77 & 15.67 \\ \(\omega_{o}=2.5\,\omega_{c}\) & 11.749 & 11.764 & 14.32 & 14.86 \\ \(\omega_{o}=3\,\omega_{c}\) & 11.394 & 11.471 & 14.16 & 14.34 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Detailed Settling Time of the Tested Method in Figure 6.  
Figure 6: Experimental results of increasing bandwidth of the LESOs. (**a**,**c**) are the performance of the \({}^{129}\)Xe, and (**b**,**d**) are the loop of \({}^{131}\)Xe. Both kinds of isotopes were driven synchronously, and the influence of the parameter \(\omega_{o}\) was specified. Consequently, the two unknown parameters are crucial to the performance of the LADRC-based magnetic field measurement. * \(\omega_{o}\) determines the frequency-observing performance significantly. * Both \(\omega_{o}\) and \(K_{c}\) affect the phase-tracking performance. Therefore, a detailed parameter tuning process shown in Figure 8 is given as follows: **Step 1:** Set \(b=\frac{K_{c}}{K_{c}-1}>0\) for a monotone nondecreasing step response. **Step 2:** Set \(\omega_{o}=K_{c}\), adjust \(K_{c}\) to make the settling time of the loop acceptable. **Step 3:** Adjust the bandwidth of LESO \(\omega_{o}\) with a fixed \(K_{c}\). Find the largest \(\omega_{o}\) based on the premise that the observer is stable.