# LMI Pole Regions for a Robust Discrete-Time Pole Placement Controller Design
## 4 Discrete-Time Robust Pole Placement Control for Magnetic Levitation (ML) Laboratory Plant  
This section presents the results obtained for a robust state feedback pole placement controller designed for a laboratory magnetic levitation plant [20], for different choices of the \(D_{R}\) region. The control aim in the ML system is to control the position of a magnetic ball levitating in the air-space. This position is controlled by the current in the coil (electromagnet) situated above the air-space. The ML can be modelled by a 3 rd order nonlinear state space system. All state variables can be measured and used for control, as follows: Ball position, ball velocity, and current in the coil. All the details for ML state space nonlinear and linearized model can be found in [21]. The sampling period for a discretized model is 1 ms. The robust state feedback pole placement controller was designed for the linearized uncertain model, Equation (1), obtained for 3 working points (WP) defined by 3 various ball positions corresponding to the state \(x_{1}\), as follows:  
\[ \begin{array}{ll}x_{10}=0.008,\\ \text{WP1}:&x_{20}=0,\\ x_{30}=0.7697\\ x_{10}=0.010,\\ \text{WP2}:&x_{20}=0,\\ x_{30}=0.9139\\ x_{10}=0.012,\\ \text{WP3}:&x_{20}=0,\\ x_{30}=1.0852\end{array} x(k+1)=\left[ \begin{array}{ccc}1.0008&0.0010&0\\ 1.6851&1.0008&-0.0233\\ 0&0&0.8300\\ 1.0008&0.0010&0\\ 1.6851&1.0008&-0.0187\\ 0&0&0.7492\\ 1.0008&0.0010&0\\ 1.6851&1.0008&-0.0146\\ 0&0&0.6391\end{array} x(k)+\left[ \begin{array}{c}0\\ -0.0098\\ 0.7479\\ 0\\ \end{array} \right]u(k)\\ \text{WP2}:&x_{20}=0,\\ x_{30}=0.9139\\ x_{10}=0.012,\\ \text{WP3}:&x_{20}=0,\\ x_{30}=1.0852\end{array}x(k+1)=\left[ \begin{array}{ccc}1.0008&0.0010&0\\ 1.6851&1.0008&-0.0146\\ 0&0.6391\end{array} x(k)+\left[ \begin{array}{c}0\\ -0.0098\\ 0.7479\\ 0\\ \end{array} \right]u(k)\\ \text{WP3}:&x_{20}=0,\\ \text{WP3}:&x_{30}=1.0852\end{array}x(k+1)=\left[ \begin{array}{ccc}1.0008&0.0 010&0\\ 1.6851&1.0008&-0.0146\\ 0&0.6391\end{array} x(k)+\left[ \begin{array}{c}0\\ -0.0154\\ 1.5878\end{array} \right]u(k)\right. \tag{16}\]  
It is important to note that the ML system is unstable and the control variable (current in the upper coil) is limited, which also limits the required pole position. The next choices of \(D_{R}\) regions and the corresponding controllers were considered and compared as follows:  
* Stabilization only: required pole region is unit circle;
* Elliptic inner approximation for 87\({}^{\circ}\) damping angle;
* New proposed ellipse-cone approximation for 50\({}^{\circ}\), 60\({}^{\circ}\), and 70\({}^{\circ}\) damping angles. For numerical reasons, in this case we considered \(R_{22}=0.01*I\) and stability degree 0.99. To track the ball position step changes, a PI controller was designed since the ML system itself does not include an integral term. The corresponding internal model principle for controller structure determination can be found, e.g., in Reference [22]. In all cases, the corresponding state feedback PI controller was designed and the corresponding closed loop poles were calculated. The designed pole placement PI controller parameters for all the considered cases are summarized in Table 1. Note that the ellipse approximation yields about nine times bigger P gain and more than 20 times bigger I gain comparing to the new proposed ellipse-cone approximation. Furthermore, the ellipse-cone approximation provides a feasible solution to Equation (8), for a 50\({}^{\circ}\) damping angle, while the elliptic approximation provides feasible solution to Equation (8) only for damping angles bigger than 86\({}^{\circ}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**D\_R Region** & **P** & **I** \\ \hline Unit circle & [106.7, 2.47, \(-0.617\)] & 0.3527 \\ Ellipse (87\({}^{\circ}\)) & [952.4, 9.75, \(-0.653\)] & 26.88 \\ Ellipse-cone (70\({}^{\circ}\)) & [134.2, 2.55, \(-0.275\)] & 1.222 \\ Ellipse-cone (60\({}^{\circ}\)) & [154.7, 2.93, \(-0.317\)] & 1.420 \\ Ellipse-cone (50\({}^{\circ}\)) & [190.7, 3.56, \(-0.368\)] & 1.831 \\ \hline \hline \end{tabular}
\end{table}