# Magnetic Levitation Control Based on Flux Density and Current Measurement
## 4 Results and Discussion  
Different tests are conducted to evaluate the plant-model mismatch, the correctness of the displacement estimation and the resolution of the sensing approach. The experimental results are obtained by selecting an air gap of 5 mm and the corresponding bias current \(i_{0}=0.7\) A to compensate the weight force. The measured coil resistance \(R=3.4\)\(\Omega\) is considered constant during the experiments and hence, a bias voltage \(u_{0}=2.21\) V is applied. As a result, there exists also a measured steady state flux density \(B_{0}=12.88\) mT which can be also corroborated from Fig 5. For safety purposes, the coil voltage is bounded to \([0,2u_{0}]\) throughout the experiments to work only in the proximity of the nominal operating point \(x_{0}\). This action avoids impulsive coil currents if the mover is manually removed. The resulting parameters of the plant model, the controller and the involved sensing coefficients at \(x_{0}\) are presented in Table 2.  
Figure 7: Control and sensing scheme. \(G_{p}\) represents the plant and \(G_{c}\) stands for the controller. The coil voltage \(u\) is manipulated to control the position of the mover. Transient responses of the system are presented in Figure 8 by varying the position reference \(q_{ref}\). The obtained results prove the feasibility of applying the indirect sensing solution and the proposed control technique. The resulting linear model is sufficient to replicate the dynamics of the plant with a slight deterioration near the overshoots. This discrepancy is expected due to the nonlinear nature of the plant. Figure 8 also presents the unfiltered behavior of current and flux density measurements. However, a low-pass filter with a cut-off frequency at 300 Hz is applied before using them for the displacement estimation. This filter is accounted for the controller synthesis. \begin{table}
\begin{tabular}{c c c c} \hline
**Subsystem** & **Parameter** & **Value** & **Unit** \\ \hline Plant Model & \(k_{q}\) & 216.30 & N/m \\  & \(k_{i}\) & 2.17 & N/A \\  & \(L_{0}\) & 41.21 & mH \\  & \(k_{p}\) & 2.17 & Vs/m \\ \hline Controller & \(K_{c}\) & 1000 & V/m \\  & \(\tau_{i}\) & 0.20 & s \\  & \(\tau_{d}\) & 0.0125 & s \\  & \(\tau_{f}\) & 0.0021 & s \\  & \(\tau_{1}\) & 0.0125 & s \\  & \(\tau_{2}\) & 0.0021 & s \\ \hline Sensing coefficients & \(k_{bi}\) & 18.4 & mT/A \\  & \(k_{bi}\) & 0.8753 & mT/mm \\ \hline \end{tabular}
\end{table} 
Table 2: Plant Model, Controller and Sensing Parameters  
Figure 8: Step response of the controlled plant for current \(i\), flux density \(B_{s}\) measurements. The displacement \(q\) is estimated from the former signals and compared to the control reference and the output of the numerical model. The correctness of the position estimates has been corroborated with a Micro-Epsilon optical displacement sensor LD1605-4 (sensitivity 5 V/mm, resolution 1 \(\upmu\)m). Figure 9 shows both, the estimates and the sensor measurements when a reference step change is applied. Similarly, Figure 10 shows a quasi-static estimation behaviour around the nominal air gap. The position estimates match the sensor measurements with enough precision. The linear approximation (10) proved sufficient to estimate the mover displacement in the vicinity of the operating point. A slight but acceptable degradation of the position estimate is advisable when the sphere moves downwards and beyond the nominal operating point. Figure 11 shows the open loop steady state behavior of the measurements from both sensors and the resulting displacement estimates. The data is obtained in a ten-second sampling window by fixing manually the mover at the nominal air gap while keeping the bias constant (i.e., the steady state condition at the operating point with the control voltage \(u\) set to \(u_{0}\)). The aim of this test is to inspect the resolution of the estimates from the measurements without the effect of the controller. The errors of the measurements are \(\pm\)6 mA and \(\pm\)69.8 \(\upmu\)T for the current and flux density sensors, respectively.