# Comparison of Nonlinear and Linear Controllers for Magnetic Levitation System
## 3 Control Design
### Robust Pole Placement--Pole Region Approach
#### 3.1.3 Robust Pole-Placement Controller Design  
Below, we recall the LMI formulation of \(D_{R}\) stability condition for closed-loop uncertain system (12) which can be directly used to compute feedback gain matrix \(K\) for robust pole placement controller, more details can be found in [27]. **Lemma 2** ([9]). : _Consider the uncertain polytopic system with state feedback control (12). If there exist positive definite matrices \(P_{i}\in R^{nxn}\) and any matrices \(H\in R^{nxn}\), \(S\in R^{mxn}\) such that the following inequality holds_  
\[ \begin{bmatrix}M_{11i}&M_{12i}\\ M_{12i}^{T}&M_{22i}\end{bmatrix} <0\quad i=1,...,N \tag{13}\]  
_where_  
\[ \begin{split} M_{11i}&=R_{11}\otimes P_{i}+R_{12}\otimes(A_{ i}H+B_{i}S)+R_{12}^{T}\otimes(A_{i}H+B_{i}S)^{T}\\ M_{22i}&=R_{22}\otimes(P_{i}-H-H^{T})\\ M_{12i}&=R_{12}^{T}\otimes(P_{i}-H^{T})+R_{22}\otimes(A_{i}H+B_{i}S), \end{split}  \tag{14}\]  
_and the state feedback controller matrix is computed as_  
\[K=SH^{-1}. \tag{15}\]  
_then the closed-loop system (10) is \(D_{R}\) stable for stability region (11)._  
Frequently, the integration term, or PI controller is required to eliminate the steady-state error of the controlled output variable. Then, the robust pole-placement controller design presented in Lemma 2 can be applied for the augmented state model which includes also the integration dynamics of PI controller. The augmented discrete-time system can be then described as  
\[A_{i_{agg}}= \begin{bmatrix}A_{i}&0\\ C&I\end{bmatrix} ,B_{i_{agg}}= \begin{bmatrix}B_{i}\\ 0\end{bmatrix} ,i=1,\ldots,N \tag{16}\]  
where matrix \(C\) corresponds to those outputs which are considered in the integration controller part. The state feedback gain matrix \(K\) is then in the following form  
\[K= \begin{bmatrix}K_{p}&K_{I}\end{bmatrix} , \tag{17}\]  
matrices \(K_{p}\) and \(K_{I}\) correspond to proportional and integral gains of the PI controller. More details can be found in [27]. For the magnetic levitation system, the ball position is required to track the reference value, therefore in this case, the system output represented by the state \(x_{1}\) enters the integration term and \(C= \begin{bmatrix}1&0&0\end{bmatrix} \). In robust-pole placement, it is very important to appropriately prescribe the required closed-loop pole region. For ML we aim at achieving the determined stability degree and relative damping. The relative damping belongs to basic closed-loop (CL) performance requirements, however, in this case, the corresponding discrete-time region is no more convex, given by a logharitmic spiral and has a \"cardioid\" shape. This nonconvex region can be advantageously approximated using convex inner approximations which then enables controller design via LMI solution. Here, the inner ellipse and angle-ellipse approximations, which we developed recently [18; 19] will be considered. The detailed description of considered pole regions is as follows. **(A) Standard Stabilizing Controller (Unit Circle)**  
In the first controller design, based on LMI approach, we will use a standard stabilizing controller from [28]:  
\[ \begin{bmatrix}H+H^{T}-P_{i}&(A_{i_{agg}}H+B_{i_{agg}}S)^{T}\\ A_{i_{agg}}H+B_{i_{agg}}S&P_{i}\end{bmatrix} \geq 0 \tag{18}\]  
\[P_{i}>0,\quad i=1,\ldots,N.\]  
the state feedback controller matrix is then computed by (15). **(B) Discrete-time Robust Pole-placement Controller Design Using Ellipse Approximation (Ellipse)**  
In this design of a discrete robust controller, we will use the \(D_{R}\) region for the ellipse, corresponding to the prescribed relative damping. Note, that the inner elliptic approximation of the originally nonconvex domain is less conservative as the circle one. The interior of the ellipse centered in \([x_{s},0]\) with semi-axes \(1/c\) and \(1/d\) for damping \(\varphi\) prescribed in continuous-time domain [29] is described by (11) with  
\[R_{11}= \begin{bmatrix}-1&-xs_{s}\\ -cx_{s}&-1\end{bmatrix} ,R_{22}= \begin{bmatrix}0&0\\ 0&0\end{bmatrix} ,R_{12}= \begin{bmatrix}0&(c-d)/2\\ (c+d)/2&0\end{bmatrix}