\section{Numerical Examples}

\label{sec:examples}

\urg{
Example content goes here.

What would be sufficient?

How much space will I have?


}



Section 4.1 of \cite{henrion2012measures} introduces a 1DOF attitude controller with linearized dynamics corresponding to a double-integrator $I \ddot{\theta} = u$ and states $x = [\theta, \dot{\theta}]$. The input $u = \textrm{sat}_L(Kx)$ is a state feedback controller $Kx$ (proportional-derivative) that saturates at levels $\pm L$. The subsystems are  linear operation $\abs{Kx} \leq L$, positive saturation $Kx \geq L$, and negative saturation $Kx \leq -L$.
These valid regions $X^k$ are separated in Fig. \ref{fig:space_all} by thin dotted diagonal lines. Maximizing $p(x) = \abs{\theta}$ where all parameters have values in Sec. 4.1.1 of \cite{henrion2012measures} is shown in Fig. \ref{fig:space}. With $\abs{\theta_0} \leq 15^\circ$ and $\abs{\dot{\theta}_0} \leq 3^\circ/\textrm{sec}$, a $d=5$ approximation finds an upper bound of $\abs{\theta_*} = 20.69^\circ$. 

The nominal moment of inertia in Fig. \ref{fig:space} is $I = 27,500 \ kg \;m^2$. Multiplicative relative uncertainty in $I$ may be introduced by replacing $I$ with $I/(1+w)$, where $w \in [-0.5, 0.5] = W$. The peak angle is raised to $\abs{\theta_*} = 51.86^\circ$ at $d=5$ at $I' = 2 I$ with this new uncertainty.


\begin{figure}[ht]
     \centering
     \begin{subfigure}[b]{0.48\linewidth}
         \centering
         \includegraphics[width=\linewidth]{img/space_const_sat.pdf}
         \caption{Certainty in $I$}
         \label{fig:space}
     \end{subfigure}
     \;
     \begin{subfigure}[b]{0.48\linewidth}
         \centering
         \includegraphics[width=\linewidth]{img/space_w_sat.pdf}
         \caption{50\% uncertainty in $I$}
         \label{fig:space_w}
     \end{subfigure}
      \caption{\label{fig:space_all} Maximum angle for 1DOF attitude controller}
\end{figure}