Section 4.1 of \cite{henrion2012measures} introduces a 1DOF attitude controller for validation of a space launcher system. These linearized dynamics corresponding to a double-integrator $I \ddot{\phi} = u$ and states $x = [\phi, \dot{\phi}]$. The input $u = \textrm{sat}_L(Kx)$ is a state feedback controller $Kx = 1000(2.475 \phi + 19.8 \dot{\phi})$ that saturates at levels $\pm L= \pm 380$. The subsystems are  linear operation $\abs{Kx} \leq L$, positive saturation $Kx \geq L$, and negative saturation $Kx \leq -L$ (deterministic switching).
These valid regions $X^k$ are separated in Fig. \ref{fig:space_all} by thin dotted diagonal lines. Maximizing $p(x) = \abs{\phi}$ (implemented as $\phi^2$) is shown in Fig. \ref{fig:space}. With $\abs{\phi_0} \leq 15^\circ$ and $\abs{\dot{\phi}_0} \leq 3^\circ/\textrm{sec}$, a degree-5 approximation finds a time-independent upper bound of $\abs{\phi_*} = 20.69^\circ$. The blue curve is the near-optimal trajectory, starting at the blue circle and extremizing $p(x)$ at the blue star.
The nominal moment of inertia in Fig. \ref{fig:space} is $I = 27,500 \ kg \;m^2$. Time-independent relative uncertainty $I$ may be introduced by replacing $I$ with $I/(1+\theta)$, where $\theta \in [-0.5, 0.5]$. The peak angle is raised to $\abs{\phi_*} = 51.86^\circ$ at $d=5$ at $I' = 2 I$ with this new uncertainty in Fig. \ref{fig:space_w}.


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