# A Randomized Algorithm for Optimal PID Controllers
## 5 Experimental Section
: _For the system from Example 1, with the same \(\Omega\) and the same \(\hat{K}^{(0)}\), the algorithm from [7] failed to find structured SOF \(\hat{K}^{(1)}\). Indeed, from another starting SOF, the above-mentioned algorithm succeeded to find a structured SOF. For_  
\[\hat{K}^{(0)}=\left[ \begin{array}{rr}\hat{K}_{1}^{(0)}&\hat{K}_{2}^{(0)}&\hat{K }_{3}^{(0)}\end{array} \right],\]  
_where_  
\[\hat{K}^{(0)}=\left[ \begin{array}{rr}-1.068321790034687&0.039576561177247&-1.1 02041006775961\\ 0.118395560901336&1.115554481033627&-0.346149598520451\\ -2.50771867599537&0.506251036541779&-3.638774839724999\end{array} \right],\]  
_and the resulting closed-loop eigenvalues are_  
\[\sigma\left(\hat{A}_{ct}\left(\hat{K}^{(best)}\right)\right)=\left\{ \begin{array} []{rr}-0.965762603991179\pm 0.955842860416991i\\ -0.507539612520429\pm 0.539780823728534i\\ -0.929257645094484\pm 0.106820410747074i\\ -0.402381802562664\pm 0.042181153576293i\end{array} \right\}.\]  
**Example 4**. : _For the system from Example 1, with the same \(\Omega\) and the same \(\hat{K}^{(0)}\), the algorithm from [7] failed to find structured SOF \(\hat{K}^{(1)}\). Indeed, from another starting SOF, the above-mentioned algorithm succeeded to find a structured SOF. For_  
\[\hat{K}^{(0)}=\left[ \begin{array}{rr}\hat{K}_{1}^{(0)}&\hat{K}_{2}^{(0)}&\hat{K }_{3}^{(0)}\end{array} \right],\]  
_where_  
\[\hat{K}^{(0)}=\left[ \begin{array}{rr}-1.068321790034687&0.039576561177247&-1.1 02041006775961\\ 0.118395560901336&1.115554481033627&-0.346149598520451\\ -2.50771867599537&0.506251036541779&-3.638774839724999\end{array} \right],\]  
_and the resulting closed-loop eigenvalues are_  
\[\sigma\left(\hat{A}_{ct}\left(\hat{K}^{(best)}\right)\right)=\left\{ \begin{array} []{rr}-0.965762603991179\pm 0.955842860416991i\\ -0.507539612520429\pm 0.539780823728534i\\ -0.929257645094484\pm 0.106820410747074i\\ -0.402381802562664\pm 0.042181153576293i\end{array} \right\}.\]  
**Example 5**. : _For the system from Example 1, with the same \(\Omega\) and the same \(\hat{K}^{(0)}\), the algorithm from [7] failed to find structured SOF \(\hat{K}^{(1)}\). Indeed, from another starting SOF, the above-mentioned algorithm succeeded to find a structured SOF. For_  
\[\hat{K}^{(0)}=\left[ \begin{array}{rr}\hat{K}_{1}^{(0)}&\hat{K}_{2}^{(0)}&\hat{K }_{3}^{(0)}\end{array} \right],\]  
_where_  
\[\hat{K}^{(0)}=\left[ \begin{array}{rr}-1.068321790034687&0.039576561177247&-1.1020410 06775961\\ 0.118395560901336&1.115554481033627&-0.346149598520451\\ -2.50771867599537&0.506251036541779&-3.638774839724999\end{array} \right],\]\[\hat{K}_{1}^{(0)} =\left[ \begin{array}{rrr}-0.001662379279286&-0.330320315815391&-0.20 4742383931633\\ 0.107146675564809&0.994156785847185&0.016601261567192\\ -0.352905474519842&-0.942806073864177&0.096372856789800\\ \end{array} \right]\] \[\hat{K}_{2}^{(0)} =\left[ \begin{array}{rrr}0.021994917878383&-0.051106382240571&0. 041069916425995\\ 0.041186023579647&0.283840911655084&0.016793398034650\\ 0.000278730967043&-0.136566568491544&0.140207270149126\\ \end{array} \right]\] \[\hat{K}_{3}^{(0)} =\left[ \begin{array}{rrr}-0.066971504986511&0.041204949529465&0. 574933673498160\\ -0.152954829804652&-0.044219575277023&0.285308855216965\\ -1.286404253117917&0.132606565745259&1.514386124380330\\ \end{array} \right],\]  
_found by the Ray-Shooting algorithm from [6], in \(CPU-time=0.0001[sec]\), applying the algorithm from [7] we have found the structured SOF_  
\[\hat{K}^{(1)}=\left[ \begin{array}{rrr}\hat{K}_{1}^{(1)}&\hat{K}_{2}^{(1)}& \hat{K}_{3}^{(1)}\end{array} \right],\]  
_where_  
\[\hat{K}_{1}^{(1)} =\left[ \begin{array}{rrr}-0.001662379279286&-0.330320315815391&-0. 204742383931633\\ 0.107146675564809&0.994156785847185&0.016601261567192\\ -0.352905474519842&-0.942806073864177&0.096372856789800\\ \end{array} \right]\] \[\hat{K}_{2}^{(1)} =\left[ \begin{array}{rrr}0.021994917878383&-0.051106382240571&0. 041069916425995\\ 0.041186023579647&0.283840911655084&0.016793398034650\\ 0.000278730967043&-0.136566568491544&0.140207270149126\\ \end{array}