\subsection{Discrete Example}


An example to demonstrate uncertain discrete peak estimation is to minimize $x_2$ on the following subsystems,
\begin{subequations}
\label{eq:poly_system}
\begin{align}
    f_1(x,w) &= \begin{bmatrix}-0.3x_1 + 0.8 x_2 + 0.1 x_1 x_2 \\ -0.75 x_1 - 0.3 x_2 +w \end{bmatrix}  \\
    f_2(x, w) &= \begin{bmatrix}0.8x_1 + 0.5 x_2 - 0.01 x_1^2 \\ -0.5 x_1 + 0.8 x_2 -0.01x_1 x_2 + w\end{bmatrix}.
\end{align}
\end{subequations}
The space under consideration is $X = [-3, 3]^2$, and the time varying uncertainty $w_t$ satisfies $w_t \in [-0.2, 0.2] = \Delta$. The valid regions for subsystems of \eqref{eq:poly_system} are $X_1 = X$ and $X_2 = X \cap (x_1 \geq 0)$. When $x_1 \geq 0$ the system may switch arbitrarily between dynamics $f_1$ and $f_2$, but when $x_2 < 0$, the system only follows dynamics $f_1$. Figure \ref{fig:poly} visualizes minimizing $x_2$ starting from the initial set $X_0 = \{x \mid (x_1 + 1.5)^2 + x_2^2 = 0.16\}$ between discrete times $t \in 0, \ldots, T$ with $T = 50$. A fourth order LMI relaxation of \eqref{eq:peak_meas_un_disc_obj} is solved aiming to maximize $p(x) = -x_2$. With $w=0$ in Fig. \ref{fig:poly_std} the bound is $P^* \leq 1.215$ ($\min x_2 \geq -1.215$), while the time varying $w$ in Fig. \ref{fig:poly_d} yields a bound of $P^* \leq 1.837$. 
\begin{figure}[ht]
     \centering
     \begin{subfigure}[b]{0.48\linewidth}
         \centering
         \includegraphics[width=\linewidth]{poly_switch_pos_std.png}
         \caption{$w_t = 0$ }
         \label{fig:poly_std}
     \end{subfigure}
     \;
     \begin{subfigure}[b]{0.48\linewidth}
         \centering
         \includegraphics[width=\linewidth]{poly_switch_pos_d.png}
         \caption{$w_t \in [-0.2, 0.2]$}
         \label{fig:poly_d}
     \end{subfigure}
      \caption{\label{fig:poly} Minimize $x_2$ on system  \eqref{eq:poly_system}}
\end{figure}

% \urg{At least one more discrete example}