# LMI Pole Regions for a Robust Discrete-Time Pole Placement Controller Design
## 3 Inner Convex Approximations to a Discrete-Time Pole Region for the Prescribed Damping
### Inner Convex Approximations to the Nonconvex Domain Corresponding to the Prescribed Damping
#### a.2.3 Right half-plane and circle  
Figure 6: Inner circle approximation of cardioid: (**a**) For a 45\({}^{\circ}\) damping angle and (**b**) for a 65\({}^{\circ}\) damping angle.  
Figure 7: Inner elliptic approximation of cardioid: (**a**) For a 45\({}^{\circ}\) damping angle and (**b**) for a 65\({}^{\circ}\) damping angle.  
This option is suitable for angles bigger than 53\({}^{\circ}\), when, for the extreme points given by Equations (14) and (15), it holds as follows: \(x_{M}-x_{0}<y_{M}\). The corresponding circle is centered in \([x_{M},0]\) and its radius is set to \(r=y_{M}\). The resulting \(D_{R}\) region is obtained as an intersection of the right half plane and the circle and the corresponding matrices are as follows:  
\(R11=\) [0 0; 0 (xM^{\prime}2-r^{\prime}2)];  
\(R12=\) [-1 0; 0 - xM];  
\(R22=\) [0 0; 0 1];  
The approximation is depicted in Figure 8a.  
\(d)\)Right half-plane and ellipse  
The ellipse is centered in \([x_{M},0]\), as in the case of b), however, its \(x\)-semi-axis is increased, so that the ellipse reaches the intersection of the cardioid with the \(y\)-axis, denoted as \(y3\). The resulting \(D_{R}\) region is obtained as an intersection of the right half plane and the ellipse and the corresponding matrices, \(R11\), \(R12\), and \(R22\), are calculated as follows:  
\(y3=exp\) (\(-pi\)/(2\({}^{*}\)tan/(f\({}^{*}\)pi/180)));  
\(ak=xM^{*}yM\)/sqrt (\(yM^{\prime}2-y3^{\prime}2\));  
\(bk=yM\);  
\(R11=\) [0 0 0; 0 - 1 - xM/ak; 0 - xM/ak - 1];  
\(R12=\) [-1 0 0; 0 0 (1/ak - 1/bk)/2; 0 (1/ak + 1/bk)/2 0];  
\(R22=\)zeros (3, 3);  
(\(\varphi\)  
The ellipse is centered in the middle of the cardioid \(x\)-axis, its \(x\)-semi-axis denoted as \(ak\) is maximized, and the \(y\)-semi-axis \(bk\) is derived so that it crosses \([x_{e},y_{e}]\), as follows:  
\(xse=(1+x0)/2\);  
\(ak=(1-x0)/2\);  
\(bk=ye^{*}ak/sqrt(ak^{2}-(xe-xse)^{*}2)\);  
and the corresponding \(D_{R}\) region matrices are as follows:  
\(R11e=[-1-xse/uk;-xse/ak-1]\);  
\(R12e=[0\ (1/ak-1/bk)/2;(1/ak+1/bk)/2\ 0]\);  
\(R22e=zeros\ (2,\ 2)\).  
The cone with the vertex in \([x_{v},0]\) and inner angle \(2\gamma\) corresponds to Figure 9 (\(\gamma\) is represented by \(ga\) in the program). The respective \(D_{R}\) region matrices for this shifted angle are as follows:  
\(ga=atan(ye/(1-xe))\);  
\(R11v=[-xv^{*}sin(ga)^{*}2\ 0;\ 0-xv^{*}sin(ga)^{*}2]\);  
\(R12v=[sin(ga)\ cos(ga);-cos(ga)\ sin(ga)]\);  
\(R22v=[0\ 0;\ 0\ 0]\).  
The \(D_{R}\) region matrices describing the intersection of the ellipse and angle can be obtained by taking \(x_{v}=1\) and merging the above matrices for the ellipse and cone, as follows:  
\(Z=zeros\ (2,\ 2)\); \(R11=[R11e\ Z;\ Z\ R11v]\);  
\(R12=[R12e\ Z;\ Z\ R12v]\);  
\(R22=[R22e\ Z;\ Z\ R22v]\).  
**Remark 2**.: _It should be noted that any \(D_{R}\) regions can be combined, so that their intersection is considered, as in the latter case, where the intersection of the ellipse and angle is described by the resulting \(D_{R}\) region matrices. \(D_{R}\) regions can be also combined with other performance criterion. Often, pole placement is considered together with \(H_{2}\) or \(H_{inf}\) minimization._  
Comparison of all the above approximations is illustrated on the example in the next section, providing the results for laboratory magnetic levitation plant.  
Figure 9: Approximation given by the intersection of the angle and ellipse (**a**) for \(x_{e}=0.7\) and (**b**) for \(x_{e}=0.6\).