# Comparison of Nonlinear and Linear Controllers for Magnetic Levitation System
## 4 State Feedback Controller for Nonlinear Magnetic Levitation System
#### 4.1.1 Stabilizing Controller Design for Magnetic Levitation (Unit Circle)  
We designed stabilizing controller via solution of LMI (18) for the augmented discrete-time system (16), where \(A_{i}\) and \(B_{i}\) are polytope vertices given by (31). This controller places the poles of the closed-loop system inside the unit circle. The resulting controller parameters are shown in Table 4. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**WP** & \(\mathbf{m}\) & \(\mathbf{a_{2 ci}}\) & \(\mathbf{x_{10}}\) & \(\mathbf{x_{20}}\) & \(\mathbf{x_{30}}\) \\ \hline WP1 & 0.0160 & \(-25.7\) & 0.010 & 0 & 0.7623 \\ WP2 & 0.0230 & \(-21.5\) & 0.010 & 0 & 0.9139 \\ WP3 & 0.0390 & \(-16.5\) & 0.010 & 0 & 1.1901 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of continuous-time model matrices (30) for 3 working points. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**WP** & \(\mathbf{m}\) & \(\mathbf{a_{21 di}}\) & \(\mathbf{a_{23 di}}\) & \(\mathbf{b_{2 di}}\) \\ \hline WP1 & 0.0160 & \(1.6851\) & \(-0.0224\) & \(-0.0149\) \\ WP2 & 0.0230 & \(1.6851\) & \(-0.0187\) & \(-0.0124\) \\ WP3 & 0.0390 & \(1.6852\) & \(-0.0143\) & \(-0.0095\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of discrete-time model matrices (31) for 3 working points. 1.2 Robust Pole-Placement Controller Design for Magnetic Levitation Using Ellipse Approximation (Ellipse)  
In this part, we design other robust pole-placement controller by a solution of LMIs (13) and (14) for the augmented discrete-time system (16), where \(A_{i}\) and \(B_{i}\) are polytope vertices given by (31). In this case, we used matrices \(R_{ii}\) and \(R_{ij}\) from (19). The results for different prescribed damping angle are shown in Table 5.  
Note, that smaller angle corresponds to higher damping, therefore more strict control requirement. For ML system, even small angle differences brings significant changes of control matrices gains. 1.3 Robust Pole-Placement Controller Design for Magnetic Levitation Using Angle-Ellipse Approximation (AE)  
Finally, we design robust pole-placement controller for more advanced AE region by solving LMIs (13) and (14) for the augmented discrete-time system (16), where \(A_{i}\) and \(B_{i}\) are polytope vertices given by (31), now we use matrices \(R_{ii}\) and \(R_{ij}\) from (20). The results for different design parameters \(x_{e}\) and \(r\) and prescribed damping angle are shown in Table 6.  
Note, that controller gains in Table 6 are significantly smaller than the ones in Table 5 though the damping angles are smaller (more strict) in the former case. This illustrates that AE provides less conservative inner approximation than Ellipse which can be noted also in simulation responses. Robust control laws (9) for standard stabilizing controller (Unit Circle), Table 4, robust pole-placement controllers obtained for Ellipse, Table 5, and Angle-Ellipse, Table 6, regions have been implemented in the simulation nonlinear model of magnetic levitation. Simulation results for step changes around the required position \(pos=0.01\) m for small, medium and big ball cases are shown in Figures 4-9. Ball position is depicted in mm for better readability. Simulation results indicate the superiority of AE controllers over UC and Ellipse ones, which is especially visible on responses for a big ball, Figures 8 and 9. The closed-loop pole regions and poles corresponding to the designed controllers are shown in Figure 10. It should be noted that AE region poles fulfilles the prescribed damping contrary to UC, while they are less strict than Ellipse ones. The closed-loop pole position influences not