## 3 The OPTCON Algorithm
### Extension of the System
\tag{46}\]  
We know that \(F_{x_{t-1}}^{\theta}=0\), \(F_{\theta_{t-1}}^{x}=0\), \(F_{\theta_{t-1}}^{\theta}=I\), and \(L_{xu}=W^{xu}\). Then we get  
\[ \begin{array}{c}\Omega_{\Psi\Psi}=\left(\left[\begin{array}{cc}(F_{x_{t-1}}^ {x})^{\prime}(I-F_{x_{t}}^{x})^{-1}&0\\ ((I-F_{x_{t}}^{x})^{-1}F_{\theta_{t}}^{x})^{\prime}&I\end{array}\right]\left[ \begin{array}{cc}K^{xx}&K^{x\theta}\\ K^{\theta x}&K^{\theta\theta}\end{array}\right]\left[\begin{array}{cc}(I-F_{x_ {t}}^{x})^{-1}F_{u_{t}}^{x}\\ 0\end{array}\right]\\ +\left[\begin{array}{cc}(F_{x_{t-1}}^{x})^{\prime}(I-F_{x_{t}}^{x})^{-1}&0 \\ ((I-F_{x_{t}}^{x})^{-1}F_{\theta_{t}}^{x})&I\end{array}\right]\left[\begin{array} []{cc}W^{xu}\\ 0\end{array}\right]\right)(\Lambda^{uu})^{-1}\\ \times\Bigg{(}\left[\begin{array}{cc}(F_{u_{t}}^{x})^{\prime}(I-F_{x_{t}}^{x })^{-1}&0\end{array}\right]\left[\begin{array}{cc}K^{xx}&K^{x\theta}\\ K^{\theta x}&K^{\theta\theta}\end{array}\right]\\ \times\Bigg{[}\begin{array}{cc}(I-F_{x_{t}}^{x})^{-1}F_{x_{t-1}}^{x}&(I-F_{x_ {t}}^{x})^{-1}F_{\theta_{t}}^{x}\\ 0\end{array}\right]\\ +\left[\begin{array}{cc}W^{ux}&0\end{array}\right]\left[\begin{array}{cc}(I-F _{x_{t}}^{x})^{-1}F_{x_{t-1}}^{x}&(I-F_{x_{t}}^{x})^{-1}F_{\theta_{t}}^{x}\\ 0&I\end{array}\right]\right).\end{array}  \tag{47}\]  
Define  
\[D=(I-F_{x_{t}}^{x})^{-1}F_{\theta_{t}}^{x}. \tag{48}\]  
Then using \(A=(I-F_{x_{t}}^{x})^{-1}F_{x_{t-1}}^{x}\) and \(B=(I-F_{x_{t}}^{x})^{-1}F_{u_{t}}^{x}\) we get the following:  
\[ \begin{array}{c}\Omega_{\Psi\Psi}=\left(\left[\begin{array}{cc}A^{\prime} &0\\ D^{\prime}&I\end{array}\right]\left[\begin{array}{cc}K^{xx}B\\ K^{\theta x}B\end{array}\right]+\left[\begin{array}{cc}A^{\prime}W^{xu}\\ D^{\prime}W^{xu}\end{array}\right]\right)(\Lambda^{uu})^{-1}\\ \times\Bigg{(}\left[\begin{array}{cc}B^{\prime}K^{xx}&BK^{\theta}\end{array} \right]\left[\begin{array}{cc}A&D\\ 0&I\end{array}\right]+\left[\begin{array}{cc}W^{ux}A\\ W^{ux}D\end{array}\right]\right)\\ =\left[\begin{array}{cc}(A^{\prime}K^{xx}B+A^{\prime}W^{xu})\\ (D^{\prime}K^{xx}B+K^{\theta x}B+D^{\prime}W^{xu})\end{array}\right](\Lambda^{uu })^{-1}\\ \times\big{[}\begin{array}{cc}(B^{\prime}K^{xx}A+W^{ux}A)&(B^{\prime}K^{xx }D+B^{\prime}K^{x\theta}+W^{ux}D)\end{array}\right]=:\left[\begin{array}{cc} \psi_{1}&\psi_{2}\\ \psi_{3}&\psi_{4}\end{array}\right].\end{array}  \tag{49}\]  
Use the definition \(\Lambda^{xu}=A^{\prime}K^{xx}B+A^{\prime}W^{xu}\) and get the following:  
\[ \begin{array}{c}\psi_{1}=\Lambda^{xu}(\Lambda^{uu})^{-1}\Lambda^{ux},\\ \psi_{2}=\Lambda^{xu}(\Lambda^{uu})^{-1}(B^{\prime}K^{xx}D+B^{\prime}K^{x\theta }+W^{ux}D),\\ \psi_{3}=(D^{\prime}K^{xx}B+K^{\theta x}B+D^{\prime}W^{ux})(\Lambda^{uu})^{-1} \Lambda^{ux},\\ \psi_{4}=(D^{\prime}K^{xx}B+K^{\theta x}B+D^{\prime}W^{ux})(\Lambda^{uu})^{-1} (B^{\prime}K^{xx}D+B^{\prime}K^{x\theta}+W^{ux}D).\end{array} \]  
For the calculation of \(H\) we need the term \(\Lambda^{\Psi\Psi}\):  
\[ \begin{array}{c}\Lambda^{\Psi\Psi}=(F_{\Psi_{t-1}})^{\prime}((I-F_{\Psi_{t}})^ {-1})^{\prime}K_{t}(I-F_{\Psi_{t}})^{-1}F_{\Psi_{t}}\\ \\ =\left(\begin{array}{cc}A^{\prime}&0\\ D^{\prime}&I\end{array}\right)\left(\begin{array}{cc}K^{xx}&K^{x\theta}\\ K^{\theta x}&K^{\theta\theta}\end{array}\right)\left(\begin{array}{cc}A&D\\ 0&I\end{array}\right)\\ =\left(\begin{array}{cc}A^{\prime}K^{xx}A&A^{\prime}(K^{xx}D+K^{x\theta})\\ (D^{\prime}K^{xx}+K^{\theta x})A&D^{\prime}(K^{xx}D+K^{x\theta})+(K^{\theta x}D+K^ {\theta\theta})\end{array}\right).\end{array}  \tag{50}\]  
Now we can calculate the term \(H\) for the extended system:  
\[H^{\Psi\Psi}=\Lambda^{\Psi\Psi}-\Lambda^{\Psi u}(\Lambda^{uu})^{-1}\Lambda^{u \Psi\Psi}=\Lambda^{\Psi\Psi}-\Omega_{\Psi\Psi}\]\[H_{t}^{\Psi\Psi}=\left( \begin{array}{cc}H^{xx}&H^{x\theta}\\ H^{\theta x}&H^{\theta\theta}\end{array}