% Hybrid Systems
\begin{frame}{Hybrid systems combine continuous dynamics with discrete mode switching}

  \begin{center}
    \begin{tikzpicture}
      \draw[thick, fill=gray!20] (0,0) rectangle (12,7);
      \node[align=center, text width=10cm] at (6,3.5) {
        \textbf{FIGURE: Hybrid System Visualization}\\[0.3cm]
        Top: Split diagram\\
        LEFT: Continuous (temp, flux, pressure curves)\\
        Equation: $\dot{x}(t) = f(x(t), q(t), u(t))$\\[0.2cm]
        RIGHT: Discrete (state machine diagram)\\
        Equation: $q(k+1) = \nu(x(k), q(k), u(k))$\\[0.3cm]
        Bottom: Reactor startup state machine\\
        Cold $\xrightarrow{T>400°F}$ Heatup $\xrightarrow{\text{near crit}}$ Crit $\rightarrow$ Power\\
        Each box shows continuous dynamics within
      };
    \end{tikzpicture}
  \end{center}

  %SPEAKER NOTES: See comments below
%
    \textbf{Nuclear plants are inherently hybrid systems}

    \textbf{Continuous Dynamics:} Reactor temperature, neutron flux, pressure, flow rates, heat transfer.
    Governed by differential equations: $\dot{x}(t) = f(x(t), q(t), u(t))$

    \textbf{Discrete Decisions:} Mode transitions, control strategy changes, safety system actuation, procedure steps.
    Governed by logic: $q(k+1) = \nu(x(k), q(k), u(k))$

    \textbf{Example:} Reactor startup: Cold Shutdown → Heatup → Approach Criticality → Low Power
    Each mode has continuous dynamics; transitions are discrete strategic decisions.
    This is exactly what human operators do today.
% (End of speaker notes)
\end{frame}