# Passivity-Based Control Design for Magnetic Levitation System
## 4 Passivity-Based Control for Magnetic Levitation System  
Magnetic levitation system are often used to verify control algorithms [18; 19; 20]. The simplified magnetic levitation control system model is shown in Figure 2[21], \(m\) is the mass of the suspended magnet, \(i\) is the current of the electromagnet coil, \(u\) is the input voltage, \(f\) is the electromagnetic levitation force, \(z\) is the distance between the electromagnet and the track, \(N\) is the number of coil windings, \(R_{0}\) is the coil resistance, \(A\) is the effective area of the magnetic pole, \(R_{T}\) is the air gap resistance, and \(\Phi_{T}\) is the main pole flux. Figure 1: Feedback control system. Neglecting the leakage flux and the magnetic reluctance of the guide rail and the magnetic core, the magnetic resistance of the magnetic circuit formed by the electromagnet and the guide rail is mainly concentrated on the air gap between the two, so the effective air gap resistance \(R_{T}\) is written as  
\[R_{T}=\frac{2z}{\mu_{0}A} \tag{13}\]  
where \(\mu_{0}\) is the permeability of vacuum, and \(\mu_{0}=4\pi\times 10^{-7}\) H/m. The flux linkage of electromagnet is written as  
\[\lambda=N\Phi_{T}=Li \tag{14}\]  
According to Equations (13) and (14), the inductance of the electromagnet winding satisfies  
\[L(z)=\frac{N\Phi_{T}}{i}=\frac{N}{i}\times\frac{Ni}{R_{T}}=\frac{\mu_{0}N^{2}A }{2z} \tag{15}\]  
According to Equation (15), the energy storage of the magnetic field is as follows  
\[E(z,i)=\frac{1}{2}L(z)i^{2}=\frac{1}{2}\frac{\mu_{0}N^{2}A}{2z}i^{2} \tag{16}\]  
Therefore, the electromagnetic force can be calculated from Equation (16) as follows (the derivative of energy versus displacement \(z\))  
\[f(z,i)=-\frac{\partial E}{\partial z}=\frac{\mu_{0}N^{2}A}{4}\bigg{[}\frac{i} {z}\bigg{]}^{2} \tag{17}\]  
Assuming that the degree of freedom of the dissipative system is \(n\), the general form of the Lagrange equation is  
\[\frac{\mathrm{d}}{\mathrm{d}t}\bigg{(}\frac{\partial L}{\partial\dot{r}_{j}} \bigg{)}-\frac{\partial L}{\partial\dot{r}_{j}}=-\frac{\partial G}{\partial \dot{r}_{j}}+Q_{j}\;j=1,\ldots\;\ldots,n \tag{18}\]  
where \(j\) is the label of the generalized coordinate, \(L\) is the Lagrange function, \(r\) and \(\dot{r}\) are the generalized coordinate and the generalized velocity, respectively, \(G\) is the dissipative function of the system, \(Q_{j}\) is a generalized and nonpotential force (except the dissipative force). In the magnetic levitation system, let \(r_{1}=z\) and \(r_{2}=q\) are generalized coordinates, and \(\dot{s}\) and \(\dot{q}=i\) are the corresponding generalized velocities. The kinetic energy of the system can be written as  
\[T=\frac{1}{2}m\dot{z}^{2}+E(z,i) \tag{19}\]  
Figure 2: Model of the electromagnetic levitation system. The potential energy can be written as  
\[V=-mgz \tag{20}\]  
Using Equations (19) and (20), we can formulate Lagrange function of the system as  
\[R_{T}=\frac{2z}{\mu_{0}A} \tag{13}\]0  
The system's dissipation function is given by  
\[R_{T}=\frac{2z}{\mu_{0}A} \tag{13}\]1  
For generalized coordinates \(r_{1}\) and \(r_{2}\), the following equations hold  
\[R_{T}=\frac{2z}{\mu_{0}A} \tag{13}\]2  
Substituting Equations (21)-(23) into Equation (18), nonlinear model of the system will be  
\[R_{T}=\frac{2z}{\mu_{0}A} \tag{13}\]3  
\[R_{T}=\frac{2z}{\mu_{0}A} \tag{13}\]4  
According to Equations (14), (15), (24), and (25), we obtain Equation (26)  
\[\sum\
olimits_{3}:\left\{ \begin{array}{l}\ddot{z}=-\frac{\lambda^{2}}{6m}+g \\ \dot{\lambda}=-\frac{2R_{0}}{k}\lambda z+u\end{array} \right. \tag{26}\]  
where \(\lambda\) is the flux linkage of electromagnet, \(k\) is a constant, and \(k=\mu_{0}AN^{2}=0.0010\ \mathrm{H}\cdot\mathrm{m}\). The nominal working point of the system is \(z_{0}=4\ \mathrm{mm}\), \(\dot{i_{0}}=3.0538\ \mathrm{A}\) and \(\lambda_{0}=0.3851\ \mathrm{V}\cdot\sec\).