## Problem Statement  
### System Dynamics  
The maglev system and single-electromagnet system can be illustrated in Figures 1 and 2. According to [PERSON]'s equation and [PERSON]'s theorem:  
\[F_{m}(i_{m}(t),x_{m}(t))=\frac{\int_{0}^{t}\psi_{m}(i_{m}(t),x_{m}(t))dt}{ \partial x_{m}(t)} \tag{1}\]  
where \(F_{m}(i_{m}(t),x_{m}(t))\) denotes electromagnetic force; \(x_{m}(t)\) represents the airgap between the electromagnet and the rail; and \(i_{m}(t)\) denotes the current. According to [PERSON] magnetic-circuit law:  
\[\psi_{m}(i_{m}(t),x_{m}(t))=N_{m}\frac{N_{m}i_{m}(t)}{R(x_{m})}, \tag{2}\]  
where \(N_{m}\) represents the coil number of turns; \(\mu_{0}\) is Air permeability; \(A_{m}\) denotes the effective magnetic pole area; and \(R(x_{m})\) can be expressed as  
\[R(x_{m})=\frac{2x_{m}(t)}{\mu_{0}A_{m}} \tag{3}\]  
Substituting Equations (2) and (3) into Equation (1), we can obtain the electromagnetic-force equation of the magnetic-levitation system as:  
\[F_{m}(i_{m}(t),x_{m}(t))=-\frac{\mu_{0}A_{m}N_{m}^{2}}{4}\bigg{[}\frac{i_{m}(t )}{x_{m}(t)}\bigg{]}^{2}. \tag{4}\]  
The relationship between voltage and current across the solenoid coil is:  
\[u_{m}(t)=i_{m}(t)R_{m}+N_{m}\dot{\phi}_{m}, \tag{5}\]  
where \(L_{m}\) is the instantaneous inductance of the electromagnet that can be expressed as:  
\[L_{m}=\frac{\mu_{0}N_{m}^{2}A_{m}}{2x_{m}(t)},\ \phi_{m}=\frac{L_{m}}{N_{m}}i_{m}(t). \tag{6}\]Finishing Equation (6) and deriving its time at both ends:  
\[\dot{\phi}_{m}=\frac{\mu_{0}N_{m}A_{m}}{2\chi_{m}(t)}\frac{di_{m}(t)}{dt}-\frac{ \mu_{0}N_{m}A_{m}i_{m}(t)}{2\chi_{m}^{2}(t)}\frac{dx_{m}(t)}{dt} \tag{7}\]  
By Equations (5) and (7), we can obtain the electrical electromagnet equation:  
\[u_{m}(t)=i_{m}(t)R_{m}+\frac{\mu_{0}N_{m}^{2}A_{m}}{2x_{m}(t)}\frac{di_{m}(t)} {dt}-\frac{\mu_{0}N_{m}^{2}A_{m}i_{m}(t)}{2x_{m}^{2}(t)}\frac{dx_{m}(t)}{dt} \tag{8}\]  
Assuming \(f_{d}\) is the disturbance force applied to the maglev-train system, the model of the magnetic-suspension system can be rewritten as follows:  
\[m\ddot{x}_{m}(t)=-\frac{\mu_{0}A_{m}N_{m}^{2}}{4}\bigg{[}\frac{i_{m}(t)}{x_{m} (t)}\bigg{]}^{2}+mg+f_{d}. \tag{9}\]  
Convert Equations (8) and (9) to state-space expressions. Here, the system state is \(x_{1}(t)=x_{m}(t)\), \(x_{2}(t)=\dot{x}_{m}(t)\), and \(y_{3}(t)=i_{m}(t)\), and the state-space expression of the floating system is:  
\[\left\{ \begin{array}{l}\dot{x}_{1}(t)=x_{2}(t)\\ \dot{x}_{2}(t)=-\frac{\mu_{0}A_{m}N_{m}^{2}}{4m}\bigg{[}\frac{x_{3}(t)}{x_{1}( t)}\bigg{]}^{2}+g+\frac{1}{m}f_{d}\\ \dot{x}_{3}(t)=\frac{x_{2}(t)x_{3}(t)}{x_{1}(t)}+\frac{2x_{3}(t)}{i_{m}0N_{m}^ {2}A_{m}}(u_{m}(t)-x_{3}(t)R_{m})\end{array} \right. \tag{10}\]  
Figure 1: Structure of maglev system. Figure 2: Single-electromagnet system. **Remark 1**. : _In an actual maglev-train levitation system, choppers are utilized to supply power to the levitation electromagnet by the principle of the current following to adjust the current for the electromagnet. As a consequence, only the current loop was considered while designing the controller. In this way, third-order dynamic equations can be degenerated into second-order nonlinear differential equations, as shown._  
\[\left\{ \begin{array}{l}\dot{x}_{1}(t)=x_{2}(t)\\ \dot{x}_{2}(t)=-\frac{\mu_{0}A_{m}N_{m}^{2}}{4m}\Big{[}\frac{i_{m}(t)}{x_{1}(t )}\Big{]}^{2}+g+\frac{1}{m}f_{d}\\ \end{array} \right. \tag{11}\]  
**Remark 2**. : _As shown in Equation (4), the relative order between nonlinear electromagnetic force \(F_{m}(t)\) and current \(i_{m}(t)\) is zero.\(\left[\frac{i_{m}(t)}{x_{1}(t)}\right]^{2}\) can be taken as \(u\) to design the controller. Since air gap \(x_{1}(t)\) can be measured at any time, it is convenient to calculate control current \(i_{m}(t)\) according to the following formula, as long as the required \(F_{m}(t)\) is determined._  
\[i_{m}(t)=\sqrt{\frac{x_{1}{}^{2}u}{\kappa}}=\sqrt{\frac{x_{1}{}^{2}F_{m}}{ \kappa}}, \tag{12}\]  
_where \(\kappa=\frac{\mu_{0}A_{m}N_{m}^{2}}{4m}\) denotes the electromagnetic-force transmission coefficient._  
**Remark 3**.