# Decoupling Control of Multi-DOF Supporting System of MLDSB
## 2 Regulating Principle of the Single-DOF Supporting System  
The adjustment principle of the radial single-DOF supporting system is shown in Figure 5. At the initial equilibrium state, the upper/lower supporting pockets adopt a constant pressure supply model and the upper/lower electromagnetic coils adopt a differential current control model. So the electromagnetic force of the upper/lower adjacent opposite poles is the same, and then the gravity of the rotor can be balanced by the hydrostatic force of the upper/lower adjacent pockets. The two adjacent magnetic poles/pockets are called the supporting unit [9].  
The single-DOF supporting system of the MLDSB is composed of two upper and lower supporting units and the rotor. The structure diagram is shown in Figure 6.  
Figure 5: Regulating principle of the radial single-DOF system.  
Figure 6: Structure diagram of the single-DOF system.  
According to the Maxwell attraction formula, the electromagnetic force can be obtained as follows [10].  
\[\left\{\begin{array}{l}f_{M}=f_{M,1}-f_{M,2}\\ f_{M,1}=2k\frac{(i_{0}-i)^{2}}{(i_{0}+2l-x\cos\theta)^{2}}\cos\theta\\ f_{M,2}=2k\frac{(i_{0}+i)^{2}}{(i_{0}+2l+x\cos\theta)^{2}}\cos\theta\\ \end{array}\right. \tag{1}\]  
where,\(f_{M,1}\) is electromagnetic force of upper pole, the unit is N. \(f_{M,2}\) is electromagnetic force of lower pole, the unit is N. \(k\) is current constant, the unit is H\(\cdot\)m, and \(k=\frac{i_{0}N^{2}A_{s}}{4}\). \(\mu_{0}\) is air permeability, the unit is H/m. \(N\) is turns of coil, the unit is dimensionless. \(A_{s}\) is magnetic's area, the unit is m\({}^{2}\). \(i_{0}\) is bias current, the unit is A. \(h_{0}\) is thickness of oil film, the unit is m. \(l\) is thickness of coating, the unit is m. \(x\) is displacement of rotor, the unit is m. \(\theta\) is angle, the unit is \({}^{\circ}\).  
According to the Navier-Stokes equation, the hydrostatic supporting force can be obtained as follows [11].  
\[\left\{\begin{array}{l}f_{L}=f_{L,2}-f_{L,1}\\ f_{L,1}=[k_{1}(u_{0}+u)-2A_{b}\dot{h}_{1}]R_{h,1}A_{c}\cos\theta\\ f_{L,2}=[k_{1}(u_{0}-u)-2A_{b}\dot{h}_{2}]R_{h,2}A_{c}\cos\theta\\ \end{array}\right. \tag{2}\]  
where,\(f_{L,1}\) is hydrostatic force of upper supporting cavity, the unit is N. \(f_{L,2}\) is hydrostatic force of lower supporting cavity, the unit is N. \(k_{1}\) is stiffness factor, the unit is L/min/V. \(u_{0}\) is flow of lower supporting cavity, the unit is m\({}^{3}\)/s. \(A_{b}\) is extrusion area, the unit is m\({}^{2}\). \(R_{h,1}\) is liquid resistance of upper supporting cavity, the unit is kg/(m\({}^{4}\cdot\)s), and \(R_{h,1}=\frac{\mu}{\overline{B}(i_{0}-x\cos\theta)^{3}}\). \(R_{h,2}\) is liquid resistance of lower supporting cavity, the unit is kg/(m\({}^{4}\cdot\)s), and \(R_{h,2}=\frac{\mu}{\overline{B}(i_{0}+x\cos\theta)^{3}}\). \(\mu\) is viscosity, the unit is Pa\(\cdot\)s. \(\overline{B}\) is flow coefficient, the unit is dimensionless. \(A_{e}\) is supporting area, the unit is m\({}^{2}\).  
The Taylor expansion of resultant force on the rotor at \(x=0\), \(\dot{x}=0\), \(i_{0}=0\), \(u_{0}=0\) is shown as follows.  
\[f_{y}=f_{M}+f_{L}=k_{x}x+k_{v}\dot{x}+k_{u}u+k_{i}i \tag{3}\]  
where, \(k_{x}\) is displacement stiffness coefficient, the unit is dimensionless. \(k_{v}\) is velocity stiffness coefficient, the unit is dimensionless. \(k_{u}\) is voltage stiffness coefficient, the unit is dimensionless. \(k_{i}\) is current stiffness coefficient, the unit is dimensionless.