## 2 Suspension Platform Structure and Control Unit Model
### Suspension Control Unit Modeling  
In the actual system, there are uncertainties, such as the fact that the track, the ground and the suspension platform cannot be completely parallel, the magnetic characteristics of each suspension control unit are different and the coupling between the units is strong, resulting in the tilt of the suspension platform at the balance position. When the system is disturbed, the variation of levitation air gap of each unit is inconsistent, which is easily leads to instability. Therefore, the decentralized control strategy is adopted in this paper to eliminate the magnetic difference between the levitation control units and reduce the coupling [21]. When establishing the model of a single suspension control unit, in order to facilitate analysis, the following assumptions are made [22; 23; 24]:  
1. There is no magnetic field coupling relationship between electromagnet and permanent magnet Halbach array;
2. The suspension control unit does not affect one another;
3. The phenomena of remanence, hysteresis and magnetic saturation are ignored. Figure 3: Suspension control unit module of PEMS platform. Figure 2: PEMS platform module. Based on the above assumptions, the dynamic equation of the suspension control unit can be expressed as:  
\[\left\{ \begin{array}{l}m\frac{d^{2}x(t)}{dt^{2}}=mg+F(i,x)\\ F(i,x)=F_{\rm e}(i,x)+F_{\rm p}(x)\\ mg=F(i_{0},x_{0})\end{array} \right. \tag{1}\]  
where \(m\) is the average mass weighed on the suspension unit affected by the suspension platform; \(g\) is the acceleration of gravity; \(i\) is the instantaneous current in the electromagnet; \(x\) is the instantaneous air gap from the guide rail to the magnetic pole surface of the electromagnet; \(i_{0}\) is the balance current in the electromagnet; \(x_{0}\) is the balance air gap and \(F(i,x)\) is combined of magnetic force of electromagnetic \(F_{\rm e}(i,x)\) and static magnetic force of the permanent magnetic \(F_{\rm p}(x)\). The electromagnetic force in the suspension control unit can be expressed as:  
\[\left\{ \begin{array}{l}F_{\rm e}(i,x)=K(\frac{i}{x})^{2}\\ K=-\frac{u_{0}\Lambda N^{2}}{4}\end{array} \right. \tag{2}\]  
where \(u_{0}\) is vacuum permeability; \(A\) is the magnetic cross-section area of iron core; \(N\) is the number of turns of the solenoid coil and \(K\) is the constant coefficient. The permanent magnet static magnetic field force in the suspension control unit can be expressed as [9]:  
\[F_{\rm p}(x)=ae^{bx}+c \tag{3}\]  
where \(a,b,c,e\) are constant coefficients \(a=3426,b=-58.26,c=202.2,e=2.71828\). From the Taylor series expansion of the magnetic field force \(F(i,x)\) in Equation (1) at the equilibrium position and ignoring the higher-order term, it can be obtained that:  
\[\left\{ \begin{array}{l}F(i,x)=F_{\rm e}(i_{0},x_{0})+F_{\rm p}(x_{0})+F^{ \prime}_{\rm e}(i_{0},x_{0})+F^{\prime}_{\rm p}(x_{0})\\ F^{\prime}_{\rm e}(i_{0},x_{0})=f^{\prime}_{\rm ei}(i_{0},x_{0})(i-i_{0})+f^{ \prime}_{\rm ex}(i_{0},x_{0})(x-x_{0})\\ F^{\prime}_{\rm p}(x_{0})=f^{\prime}_{\rm px}(x_{0})(x-x_{0})\end{array} \right. \tag{4}\]  
and \(f^{\prime}_{\rm ei}(i_{0},x_{0}),f^{\prime}_{\rm ex}(i_{0},x_{0}),f^{\prime}_{ \rm pi}(x_{0})\) can be obtained from the differential partial derivative calculation:  
\[\left\{ \begin{array}{l}f^{\prime}_{\rm ei}(i_{0},x_{0})=\left(\frac{\partial F _{\rm e}(i,x)}{\partial t}\right)\left|i=i_{0},x=x_{0}\right.\right)=2K\frac {i_{0}}{x_{0}^{2}}=k_{\rm ei}\\ f^{\prime}_{\rm ex}(i_{0},x_{0})=\left(\frac{\partial F_{\rm e}(i,x)}{\partial x }\right|i=i_{0},x=x_{0})=-2K\frac{i_{0}^{2}}{x_{0}^{2}}=k_{\rm ex}\\ f^{\prime}_{\rm px}(x_{0})=\frac{dF_{\rm p}(x)}{dx}=abe^{bx_{0}}=k_{\rm px} \end{array} \right. \tag{5}\]  
where \(k_{\rm ei},k_{\rm ex},k_{\rm px}\) are the computed results of \(f^{\prime}_{\rm ei}(i_{0},x_{0}),f^{\prime}_{\rm ex}(i_{0},x_{0}),f^{\prime} _{\rm pi}(x_{0})\).