## 3 Magnetic Levitation Control System
### Feedback Controller Design  
Assuming that the control bandwidth of the current controller is sufficiently fast compared to the magnetic levitation control bandwidth, the sensor noise is small, and the fluctuation of the 5-DOF motion is not large so that the nonlinear term of the dynamic  
Figure 5: Block diagram of the feedback control system of the magnetic levitation wafer transfer system. equation can be ignored, and the block diagram can be simplified as shown in Figure 6. Furthermore, the magnetic levitation feedback controller is designed as follows:  
\[\mathbf{u}=\mathbf{C}(\mathbf{q_{ref}}-\mathbf{q}) \tag{9}\]  
where  
\[\mathbf{C}=diag(C_{x}\quad C_{z}\quad C_{\theta_{x}}\quad C_{\theta_{y}}\quad C_ {\theta_{z}})\]  
A key feature of this controller structure is that there is no coupling term between the DOFs, so it can easily implement independent control characteristics for each DOF. In this study, a proportional-integral-derivative (PID) with a low-pass filter control structure is used. By combining the dynamic equation Equation (4) and the feedback controller Equation (9), this model can be expressed as follows by utilizing the Laplace transform. \[\ddot{\mathbf{q}}=\mathbf{M}^{-1}\mathbf{C}(\mathbf{q_{ref}}-\mathbf{q})+ \mathbf{M}^{-1}\mathbf{d}\longrightarrow s^{2}\mathbf{Q}=\mathbf{M}^{-1} \mathbf{C}(\mathbf{Q_{ref}}-\mathbf{Q})+\mathbf{M}^{-1}\mathbf{D} \tag{10}\]  
The above equation can be expressed as  
\[\mathbf{Q}=[s^{2}\mathbf{I}+\mathbf{M}^{-1}\mathbf{C}]^{-1}\mathbf{M}^{-1} \mathbf{C}\mathbf{Q_{ref}}+[s^{2}\mathbf{I}+\mathbf{M}^{-1}\mathbf{C}]^{-1} \mathbf{M}^{-1}\mathbf{D} \tag{11}\]  
where  
\[C(s)=k_{p}+k_{i}\frac{1}{s}+k_{d}\frac{Ns}{s+N}\]  
Equation (11) can be used to obtain the transfer functions for each DOF. For example, the transfer function for the \(x\)-axis is  
\[x=\frac{\#2}{\#1}x_{ref}+\frac{\#3}{\#1}F_{dx} \tag{12}\]  
where  
\[ \begin{array}{l}\#1=ms^{4}+mNs^{3}+(k_{p}+k_{d}N)s^{2}+(k_{p}N+k_{i})s+k_{i}N \\ \#2=(k_{p}+k_{d}N)s^{2}+(k_{p}N+k_{i})s+k_{i}N\\ \#3=s^{3}+Ns\end{array} \]  
The equation labeled #1 is called the characteristic equation, which reflects various feedback control characteristics such as the stability and response. In this study, the controller is designed to achieve a control bandwidth of approximately 10 Hz, a phase margin of more than 30 degrees, and a gain margin as large as possible. The feedback controller is designed to meet the desired system requirements by adjusting the control gain using the loop shaping technique. Loop shaping is a well-known systematic control system design method that can be used to achieve the desired control performance, stability, and robustness by shaping the loop transfer function of the control system [24]. In the low-frequency range, the magnitude of loop gain is closely related to control performance in terms of external disturbance attenuation and tracking performance. In the high-frequency range, the magnitude of loop gain is closely related to control performance in terms of sensor noise attenuation. Figure 6: Simplified block diagram of feedback control for system identification. After completing the controller design on a MATLAB simulation, a sine sweep test was performed to verify that the developed feedback controller works properly in the actual physical system. The sine sweep test is an experiment that analyzes the frequency response of the developed feedback control system by examining the relationship between the input and output. In the sine sweep test, a chirp signal is added to the reference. The amplitude of the chirp signal applied was \(\pm\)0.01 mm, and the frequency was set to gradually change from 0.1 Hz to 100 Hz over 300 s. During the test, the reference, error, and output were measured. Figure 7 shows the reference and output measured by conducting a sine sweep test on the x-axis. Figure 8 shows the frequency response characteristics based on the measured reference, error, and output estimated by the MATLAB System Identification Toolbox.