## 3 The OPTCON Algorithm
### Extension of the System  
Recall that the algorithm deals with a quadratic criterion function when the parameters of the system equations are unknown. One technique to incorporate the uncertain parameters is to treat the random parameters as additional state variables. Thus, apply the equations of the objective function to the system  
\[\Psi_{t}=\left[ \begin{array}{c}x_{t}\\ \theta_{t}\end{array} \right].\]  
Then  
\[L_{t}(\Psi_{t},u_{t})=\tfrac{1}{2}[(x_{t}-\widetilde{x}_{t})^{ \prime}\vdots 0]\left[ \begin{array}{ccc}W^{xx}&\vdots&0\\ \cdots&&\cdots\\ 0&\vdots&0\end{array} \right]\left[ \begin{array}{c}(x_{t}-\widetilde{x}_{t}) \\ \cdots\\ 0\end{array} \right] \tag{42}\] \[+[(x_{t}-\widetilde{x}_{t})^{\prime}\vdots 0]\left[ \begin{array}{c} W^{xu}\\ \cdots\\ 0\end{array} \right][u_{t}-\widetilde{u}_{t}]+\tfrac{1}{2}[u_{t}-\widetilde{u}_{ t}]^{\prime}W^{uu}[u_{t}-\widetilde{u}_{t}]\]  
and  
\[f_{t}(\Psi_{t},u_{t})=\left[ \begin{array}{c}f^{x}(\Psi_{t},u_{t})\\ f^{\theta}(\Psi_{t},u_{t})\end{array} \right]=\left[ \begin{array}{c}f(x_{t},x _{t-1},u_{t},\theta_{t})\\ 1\theta_{t-1}\end{array} \right]. \tag{43}\]  
In order to calculate the adjusted components of the objective function (especially (40) and (41)) we need the terms \(H\) and \(\Omega\). First, we calculate the adapted \(\Omega_{\Psi\Psi}\) using Definition (30):  
\[\Omega_{\Psi\Psi} =\Lambda^{\Psi u}(\Lambda^{uu})^{-1}\Lambda^{u\Psi}=[F_{\Psi_{t-1 }}((I-F_{\Psi_{t}})^{-1})^{\prime}K_{t}(I-F_{\Psi_{t}})^{-1}F_{u} \tag{44}\] \[+F_{\Psi_{t-1}}((I-F_{\Psi_{t}})^{-1})^{\prime}L_{\Psi u}]( \Lambda^{uu})^{-1}[F_{u}((I-F_{\Psi_{t}})^{-1})^{\prime}K_{t}(I-F_{\Psi_{t} })^{-1}F_{\Psi_{t-1}}\] \[+L_{\Psi u}((I-F_{\Psi_{t}})^{-1})^{\prime}F_{\Psi_{t-1}}]\] \[=\left(\left[ \begin{array}{cc}(F_{\Psi_{t-1}}^{x})^{\prime}&F_{ \Psi_{t-1}}^{\theta}\\ (F_{\Psi_{t-1}}^{x})^{\prime}&F_{\Psi_{t-1}}^{\theta}\end{array} \right]\left[ \begin{array}{cc}((I-F_{\Psi_{t}}^{x})^{-1})^{\prime}&0\\ ((I-F_{\Psi_{t}}^{x})^{-1}F_{\Psi_{t}}^{x})^{\prime}&I\end{array} \right] \left[ \begin{array}{cc}K^{xx}&K^{\alpha\theta}\\ K^{\theta x}&K^{\theta\theta}\end{array} \right]\] \[\times\left[ \begin{array}{cc}((I-F_{\Psi_{t}}^{x})^{-1})^{ \prime}&(I-F_{\Psi_{t}}^{x})^{-1}F_{\theta_{t}}^{x}\\ 0&I\end{array} \right]\left[ \begin{array}{cc}F_{u}^{x}\\ F_{\Psi_{t}}^{x}\end{array} \right]+\left[ \begin{array}{cc}F_{\Psi_{t-1}}^{x} &F_{\Psi_{t-1}}^{\theta}\\ F_{\Psi_{t-1}}^{x}&F_{\Psi_{t-1}}^{\theta}\end{array} \right]\] \[\times\left[ \begin{array}{cc}(I-F_{\Psi_{t}}^{x})^{-1}&0\\ (I-F_{\Psi_{t}}^{x})^{-1}F_{\theta_{t}}^{x}&I\end{array} \right]\left[ \begin{array} []{cc}L_{xu}\\ 0\end{array} \right]\right)(\Lambda^{uu})^{-1}\bigg{(}\left[ \begin{array}{cc}F_{ u}^{x}&0\\ (I-F_{\Psi_{t}}^{x})^{-1}F_{\theta_{t}}^{x}&I\end{array} \right]\] \[\times\left[ \begin{array}{cc}K^{xx}&K^{\alpha\theta}\\ K^{\theta x}&K^{\alpha\theta}\end{array} \right]\left[ \begin{array}{cc}(I-F_{ \Psi_{t}}^{x})^{-1}&(I-F_{\Psi_{t}}^{x})^{-1}F_{\theta_{t}}^{x}\\ 0&I\end{array} \right]\left[ \begin{array}{cc}F_{\Psi_{t-1}}^{x}&F_{\Psi_{t-1}}^{ \theta}\\ F_{\Psi_{t-1}}^{\theta}&F_{\theta_{t-1}}^{\theta}\end{array} \right]\] \[+\left[ \begin{array}{cc}L_{ux}&0\end{array} \right]\left[ \begin{array}{cc}(I-F_{\Psi_{t}}^{x})^{-1}&(I-F_{\Psi_{t}}^{x})^{-1}F_{ \theta_{t}}^{x}\\ 0&I\end{array} \right]\left[ \begin{array}{cc}F_{\Psi_{t-1}}^{x}&F_{\Psi_{t-1}}^{ x}\\ F_{\Psi_{t-1}}^{\theta}&F_{\theta_{t-1}}^{\theta}\end{array} \right]\bigg{)},\]  
where  
\[I-F_{\Psi_{t}} =\left( \begin{array}{cc}I&0\\ 0&I\end{array} \right)-\left( \begin{array}{cc}F_{x_{t}}^{x}&F_{\theta_{t}}^{ x}\\ F_{\Psi_{t}}^{\theta}&F_{\theta_{t}}^{\theta}\end{array} \right)=\left( \begin{array}{ cc}I&0\\ 0&I\end{array} \right)-\left( \begin{array}{cc}F_{x_{t}}^{x}&F_{\theta_{t}}^{x}\\ 0&0\end{array} \right) \tag{45}\] \[=\left( \begin{array}{cc}I-F_{x_{t}}^{x}&(-F_{\Psi_{t}}^{x})\\ 0&I\end{array} \right)\]and  
\[(I-F_{\Psi_{t}})^{-1}=\left( \begin{array}{cc}(I-F_{\Psi_{t}}^{x})^{-1}&(I-F_{x_{ t}}^{x})^{-1}F_{\theta_{t}}^{x}\\ 0&I\end{array} \right).