## Appendix A Derivation of the Matrices \(\mathbf{D}_{0}\) and \(\mathbf{D}_{1}\) in (24)  
This Appendix A provides an element-by-element derivation of the matrices \(\mathbf{D}_{0}\) and \(\mathbf{D}_{1}\) in (24). The first constraint (right bound for the eigenvalues' real part) on the area of admissible eigenvalues is given by  
\[\Re\{s\}\leq-\gamma\enspace. \tag{11}\]  
It is equivalent to  
\[\frac{1}{2}\cdot(s+\bar{s})\leq-\gamma\quad\Longrightarrow\quad 2\gamma+s+ \bar{s}\leq 0\enspace. \tag{12}\]  
Analogously, the second constraint (left bound for the eigenvalues' real part) is given by  
\[\Re\{s\}\geq-\delta\enspace. \tag{13}\]  
It is equivalent to  
\[\frac{1}{2}\cdot(s+\bar{s})\geq-\delta\quad\Longrightarrow\quad 2\delta+s+ \bar{s}\geq 0\quad\Longrightarrow\quad-2\delta-s-\bar{s}\leq 0\enspace. \tag{14}\]  
Finally, the damping sector with the angle \(\theta\), \(0\leq\theta<\frac{\pi}{2}\), corresponds to the constraint  
\[\Re\{s\}\cdot\tan\theta\leq-\Im\{s\}\quad\Longrightarrow\quad\Re\{s\}\cdot \sin\theta\leq-\Im\{s\}\cdot\cos\theta\enspace. \tag{15}\]  
Due to the negativity of both sides, squaring the second inequality in (15) yields  
\[\Re^{2}\{s\}\cdot\sin^{2}\theta\geq\Im^{2}\{s\}\cdot\cos^{2}\theta\quad\text {with}\quad\Re\{s\}\cdot\sin\theta\leq 0\enspace, \tag{16}\]  
which is equivalent to  
\[\left(\frac{s+\bar{s}}{2}\right)^{2}\cdot\sin^{2}\theta\geq\left(\frac{s-\bar{ s}}{2j}\right)^{2}\cdot\cos^{2}\theta\quad\text{with}\quad\left(\frac{s+\bar{s}}{2} \right)\cdot\sin\theta\leq 0 \tag{17}\]  
with \(j^{2}=-1\) and, hence, also  
\[\left(s+\bar{s}\right)^{2}\cdot\sin^{2}\theta+\left(s-\bar{s}\right)^{2}\cdot \cos^{2}\theta\geq 0\quad\text{with}\quad\left(s+\bar{s}\right)\cdot\sin \theta\leq 0\enspace. \tag{18}\]Combining both scalar inequalities into a joint matrix inequality with the constraints of negative definiteness yields (by accounting for [PERSON]'s criterion)  
\[ \begin{bmatrix}(s+\bar{s})\cdot\sin\theta&(s-\bar{s})\cdot\cos\theta\\ -(s-\bar{s})\cdot\cos\theta&(s+\bar{s})\cdot\sin\theta\end{bmatrix} =s\cdot \begin{bmatrix}\sin\theta&\cos\theta\\ -\cos\theta&\sin\theta\end{bmatrix} +\bar{s}\cdot \begin{bmatrix}\sin\theta&- \cos\theta\\ \cos\theta&\sin\theta\end{bmatrix} \preceq 0\enspace.\] (A9)  
Now, a combination of all inequalities (A2), (A4), and (A9) in the form  
\[\underbrace{\left[ \begin{array}{cccc}2\gamma&0&0&0\\ 0&1&0&0\\ 0&1&0&0&0\end{array} \right]}_{\mathbf{D}_{0}}+s\cdot\underbrace{\left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0&0\end{array} \right]}_{\mathbf{D}_{1}}+s\cdot\underbrace{\left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0&0\end{array} \right]}_{\mathbf{D}_{1}}+s\cdot\mathbf{D}_{1}^{T}\preceq 0\] (A10)  
completes the derivation of the matrices listed in (24). **Remark A1**. : _In contrast to the following Appendix B, the inequalities above are formulated as semi-definiteness constraints to allow for the case \(\theta=0\) considered in the application scenario._  
Appendix B Generalization of (13) toward the Consideration of the Eigenvalue Region Constraints of the Output Feedback Controller  
To account for the consideration of eigenvalue constraints in the form  
\[\mathbf{F}_{\Gamma,\mathcal{C}}(s):=\mathbf{D}_{0,\mathcal{C}}+s\cdot\mathbf{ D}_{1,\mathcal{C}}+\bar{s}\cdot\mathbf{D}_{1,\mathcal{C}}^{T}\prec 0\] (A11)  
with the given matrices \(\mathbf{D}_{0,\mathcal{C}}\) and \(\mathbf{D}_{1,\mathcal{C}}\) in the sense of a generalization of Corollary 1, the inequality constraint (11) is first replaced by  
\[\mathbf{D}_{0,\mathcal{C}}\otimes\mathbf{P}+\mathbf{D}_{1,\mathcal{C}}\otimes \left(\left(\mathbf{A}_{v}-\mathbf{B}_{v}\mathbf{K}_{y}\mathbf{C}_{y}^{+} \bar{\mathcal{C}}\right)^{T}\cdot\mathbf{P}\right)+\mathbf{D}_{1,\mathcal{C}} ^{T}\otimes\left(\mathbf{P}\cdot\left(\mathbf{A}_{v}-\mathbf{B}_{v}\mathbf{K} _{y}\mathbf{C}_{y}^{+}\bar{\mathcal{C}}\right)\right)\prec 0\] (A12)  
for each polytope vertex \(v\).