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D  Presentations/ERLM/bouncing_ball_hybrid.png

A  Presentations/ERLM/images/4_research_approach/phase_portrait_sg1.png

A  Presentations/ERLM/images/4_research_approach/two_loop.png

A  Presentations/ERLM/images/7_broader_impacts/billion.jpg

M  Presentations/ERLM/main.aux

M  Presentations/ERLM/main.fdb_latexmk

M  Presentations/ERLM/main.fls

M  Presentations/ERLM/main.log
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@ -60,9 +60,10 @@
\input{slides/3_Limitations_v2}
\input{slides/4_Research_Approach_v2}
\input{slides/5_Metrics_of_Success_v2}
\input{slides/6_Risks_and_Contingencies_v2}
\input{slides/7_Broader_Impacts_v2}
\input{slides/8_Money_Slide_v2}
%Supplemenatry
\input{slides/6_Risks_and_Contingencies_v2}
\end{document}

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Presentations/ERLM/slides/4_Research_Approach_v2.tex
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\node[draw=textcolor, fill=primary!30!red, right=0.1cm of chess, yshift=0.4cm] {Controller};
\end{tikzpicture}
\end{column}
\begin{column}{0.5\textwidth}
\pause
\begin{tikzpicture}[
state/.style={circle, draw=textcolor, fill=secondary, minimum size=1.2cm},
transition/.style={->, >=stealth, thick}
@ -135,52 +135,15 @@
\end{frame}
\begin{frame}{Finally, we will build continuous controllers to move between
discrete states}
\begin{frame}{Finally, we will build continuous controllers with formal methods
to ensure transitions between modes}
\begin{columns}
\begin{column}{0.5\textwidth}
\includegraphics[height=0.7\textheight]{bouncing_ball_hybrid.png}
\begin{column}{0.4\textwidth}
\includegraphics[height=0.9\textwidth]{images/4_research_approach/two_loop.png}
\end{column}
\begin{column}{0.5\textwidth}
\only<2>{
\begin{alertblock}{Key Challenge}
\small
Verify continuous control behavior across discrete mode transitions
\end{alertblock}
}
\only<3->{
\begin{block}{Key Challenge}
\small
Verify continuous control behavior across discrete mode transitions
\end{block}
}
\vspace{0.3cm}
\only<3>{
\begin{alertblock}{Reachable Set}
\small
$\mathcal{R}(t) = \{x(t) \mid x(0) \in X_0, \dot{x} = f(x)\}$
\end{alertblock}
}
\only<4->{
\begin{block}{Reachable Set}
\small
$\mathcal{R}(t) = \{x(t) \mid x(0) \in X_0, \dot{x} = f(x)\}$
\end{block}
}
\vspace{0.3cm}
\only<4>{
\begin{alertblock}{Barrier Certificate}
\small
$B(x) > 0 \land \nabla B \cdot f(x) \leq 0 \implies x \in \text{Safe}$
\end{alertblock}
}
\begin{column}{0.6\textwidth}
\includegraphics[height=0.75\textheight]{images/4_research_approach/phase_portrait_sg1.png}
\end{column}
\end{columns}

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Presentations/ERLM/slides/5_Metrics_of_Success_v2.tex
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@ -1,19 +1,16 @@
\begin{frame}{Success will be measured through Technology Readiness Level advancement}
\begin{frame}{Success will be measured through Technology Readiness Level (TRL) advancement}
\begin{columns}
\begin{column}{0.3\textwidth}
\begin{block}{Why TRLs?}
\small
Bridge gap between proof-of-concept and deployment
\begin{itemize}
\item[] Measure both rigor and feasibility
\end{itemize}
\begin{block}{TRL Goal}
\textbf{Current:} TRL 2-3\\
\textbf{Target:} TRL 5
\end{block}
\vspace{0.5cm}
\textbf{Current:} TRL 2-3\\
\textbf{Target:} TRL 5
\end{column}
\begin{column}{0.7\textwidth}

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Presentations/ERLM/slides/6_Risks_and_Contingencies.tex
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\begin{frame}{Four primary risks are identified with clear mitigation and contingency plans}
\begin{enumerate}
\item<1-> \alert<1>{Computational Tractability of Synthesis}
\item \alert{Computational Tractability of Synthesis}
\item<2-> \alert<2>{Discrete-Continuous Interface Complexity}
\item \alert{Discrete-Continuous Interface Complexity}
\item<3-> \alert<3>{Procedure Formalization Completeness}
\item \alert{Procedure Formalization Completeness}
\item<4-> \alert<4>{Hardware-in-the-Loop Integration}
\item \alert{Hardware-in-the-Loop Integration}
\end{enumerate}
\end{frame}

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Presentations/ERLM/slides/7_Broader_Impacts_v2.tex
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@ -1,75 +1,64 @@
\begin{frame}{Broader Impact: Multi-billion dollar O\&M cost reduction}
\begin{frame}{Reducing operations costs for new nuclear power is a billion
dollar problem}
\centering
\begin{block}{The Economic Opportunity}
\small
Datacenter electricity demand projected to reach \textbf{1,050 TWh/year by 2030}
\end{block}
\only<1>{
\includegraphics[width=0.8\textwidth]{images/7_broader_impacts/billion.jpg}
}
\vspace{0.5cm}
\only<2>{
\begin{tikzpicture}
% Adjustable parameters
\def\iconsize{1.6cm}
\def\xspacing{1.7cm}
\def\yspacing{1.0cm}
\textbf{If supplied by nuclear power:}
\begin{align*}
\text{Total annual cost} &= 1,050 \text{ TWh/yr} \times \$88.24/\text{MWh} \\
&= \textbf{\$92.7 billion/year}
\end{align*}
\pause
\vspace{0.3cm}
\textbf{O\&M represents 23-30\% of LCOE:}
\begin{align*}
\text{O\&M costs} &= \$92.7\text{B} \times 0.23\text{-}0.30 \\
&= \textbf{\$21-28 billion/year}
\end{align*}
\pause
\vspace{0.5cm}
\begin{center}
\large
\textbf{Autonomous control directly addresses\\
this multi-billion dollar challenge}
\end{center}
% 12 Senior Operators (2 per shift <20> 6 shifts)
\foreach \row in {0,...,5} {
\foreach \col in {0,...,9} {
\node at (\col*\xspacing, -\row*\yspacing)
{\includegraphics[width=\iconsize]{images/7_broader_impacts/billion.jpg}
};
}
}
\end{tikzpicture}
}
\end{frame}
\begin{frame}{Beyond nuclear: A generalizable framework for safety-critical autonomy}
\begin{columns}
\begin{column}{0.5\textwidth}
\begin{block}{Why Nuclear First?}
\small
\begin{itemize}
\item Highest regulatory requirements
\item Most safety-critical domain
\item Procedures already documented
\item Establishes regulatory pathway
\end{itemize}
\end{block}
\end{column}
\begin{column}{0.5\textwidth}
\begin{block}{Future Applications}
\small
\begin{itemize}
\item Chemical process control
\item Aerospace systems
\item Autonomous transportation
\item Critical infrastructure
\end{itemize}
\end{block}
\end{column}
\end{columns}
\vspace{1cm}
\begin{center}
\textbf{Translate procedures $\rightarrow$ Synthesize logic $\rightarrow$ Verify behavior}\\
\small Applicable to any hybrid system with documented operational requirements
\end{center}
\end{frame}
% \begin{frame}{Beyond nuclear: A generalizable framework for safety-critical autonomy}
%
% \begin{columns}
% \begin{column}{0.5\textwidth}
% \begin{block}{Why Nuclear First?}
% \small
% \begin{itemize}
% \item Highest regulatory requirements
% \item Most safety-critical domain
% \item Procedures already documented
% \item Establishes regulatory pathway
% \end{itemize}
% \end{block}
% \end{column}
%
% \begin{column}{0.5\textwidth}
% \begin{block}{Future Applications}
% \small
% \begin{itemize}
% \item Chemical process control
% \item Aerospace systems
% \item Autonomous transportation
% \item Critical infrastructure
% \end{itemize}
% \end{block}
% \end{column}
% \end{columns}
%
% \vspace{1cm}
%
% \begin{center}
% \textbf{Translate procedures $\rightarrow$ Synthesize logic $\rightarrow$ Verify behavior}\\
% \small Applicable to any hybrid system with documented operational requirements
% \end{center}
%
% \end{frame}

2
Presentations/ERLM/theme/beamerthemedane_native.sty
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@ -12,7 +12,7 @@
% REQUIREMENTS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\RequirePackage{tikz}
\RequirePackage[dvipsnames]{xcolor}
\RequirePackage{xcolor}
\RequirePackage{graphicx}
\RequirePackage{pgfplots}
\RequirePackage{geometry}

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using DifferentialEquations
using Plots
"""
Two-Loop Reactor System - FINE Parameter Sweep over μ
Using ONLY constants from Python script
"""
println("=== Two-Loop Reactor: Fine μ Sweep ===\n")
# Constants from Python script
const C_0 = 33.33 # Base Heat Capacity [%-sec/°F]
const τ_0 = 0.75 * C_0 # Base Time Constant [sec]
# Initial conditions from Python script
const T_initial = 450.0 # All temperatures start at 450°F
# Power levels from Python script
const P_r = 100.0 # Reactor power
const Q_sg = 50.0 # Heat removal per steam generator (equal loading)
println("=== Parameters (from Python script) ===")
println("C_0 = $C_0 %-sec/°F")
println("τ_0 = $τ_0 sec")
println("P_r = $P_r")
println("Q_sg = $Q_sg per SG")
# Two-loop system ODEs
function two_loop!(du, u, p, t)
T_hot, T_cold1, T_cold2 = u
μ = p[1]
# System parameters (from Python script)
C_r = μ * C_0
C_sg = (1 - μ) * C_0 / 2
W = C_0 / (2 * τ_0)
# Energy balances (exactly from Python script)
# C_r * dT_hot/dt = P_r - W*(T_hot - T_cold1) - W*(T_hot - T_cold2)
du[1] = (P_r - W * (T_hot - T_cold1) - W * (T_hot - T_cold2)) / C_r
# C_sg * dT_cold1/dt = W*(T_hot - T_cold1) - Q1
du[2] = (W * (T_hot - T_cold1) - Q_sg) / C_sg
# C_sg * dT_cold2/dt = W*(T_hot - T_cold2) - Q2
du[3] = (W * (T_hot - T_cold2) - Q_sg) / C_sg
return du
end
# Initial conditions (exact, from Python script)
u0 = [T_initial, T_initial, T_initial]
println("\n=== Initial Conditions ===")
println("T_hot_0 = $T_initial °F")
println("T_cold1_0 = $T_initial °F")
println("T_cold2_0 = $T_initial °F")
# Fine μ sweep
μ_values = range(0.5, 0.75, length=50) # VERY FINE sweep
println("\n=== μ Sweep ===")
println("μ range: [0.5, 0.75]")
println("Number of cases: $(length(μ_values))")
# Time span
tspan = (0.0, 30.0)
t_save = range(0, 30, length=300)
# Solve for all μ values
println("\nSolving $(length(μ_values)) cases...")
all_sols = []
for (i, μ) in enumerate(μ_values)
if i % 10 == 0
print(".")
end
p = [μ]
prob = ODEProblem(two_loop!, u0, tspan, p)
sol = solve(prob, Tsit5(), saveat=t_save)
push!(all_sols, sol)
end
println(" Done!")
# Extract data
times = all_sols[1].t
all_T_hot = [[sol.u[i][1] for i in 1:length(sol)] for sol in all_sols]
all_T_cold1 = [[sol.u[i][2] for i in 1:length(sol)] for sol in all_sols]
all_T_cold2 = [[sol.u[i][3] for i in 1:length(sol)] for sol in all_sols]
# Compute T_ave for each μ
all_T_ave = []
for (i, μ) in enumerate(μ_values)
T_ave = μ * all_T_hot[i] .+ (1 - μ) * (all_T_cold1[i] .+ all_T_cold2[i]) ./ 2
push!(all_T_ave, T_ave)
end
# Compute envelopes
T_hot_min = [minimum([all_T_hot[i][j] for i in 1:length(μ_values)]) for j in 1:length(times)]
T_hot_max = [maximum([all_T_hot[i][j] for i in 1:length(μ_values)]) for j in 1:length(times)]
T_ave_min = [minimum([all_T_ave[i][j] for i in 1:length(μ_values)]) for j in 1:length(times)]
T_ave_max = [maximum([all_T_ave[i][j] for i in 1:length(μ_values)]) for j in 1:length(times)]
println("\n=== Creating Plots ===")
# Plot 1: T_hot reach tube
p1 = plot(times, T_hot_min, fillrange=T_hot_max,
xlabel="Time (s)", ylabel="T_hot (°F)",
title="Hot Leg Temperature Reach Tube",
fillalpha=0.4, color=:red, lw=2,
label="Reach tube", legend=:bottomright)
# Plot 2: T_ave reach tube
p2 = plot(times, T_ave_min, fillrange=T_ave_max,
xlabel="Time (s)", ylabel="T_avg (°F)",
title="Average Temperature Reach Tube",
fillalpha=0.4, color=:orange, lw=2,
label="Reach tube", legend=:bottomright)
# Plot 3: Phase portrait - just the reach tube boundary
p3 = plot(xlabel="T_avg (°F)", ylabel="T_hot (°F)",
title="Phase Portrait: T_hot vs T_avg",
legend=:bottomright, size=(700, 700))
# Plot the envelope
plot!(p3, [T_ave_min; reverse(T_ave_max)], [T_hot_min; reverse(T_hot_max)],
seriestype=:shape, fillalpha=0.3, color=:purple,
label="Reachable region", lw=2)
# Add boundary traces
plot!(p3, T_ave_min, T_hot_min, color=:blue, lw=2, label="μ = 0.5")
plot!(p3, T_ave_max, T_hot_max, color=:red, lw=2, label="μ = 0.75")
# Plot 4: ΔT reach tube
ΔT_min = [minimum([all_T_hot[i][j] - all_T_cold1[i][j] for i in 1:length(μ_values)]) for j in 1:length(times)]
ΔT_max = [maximum([all_T_hot[i][j] - all_T_cold1[i][j] for i in 1:length(μ_values)]) for j in 1:length(times)]
p4 = plot(times, ΔT_min, fillrange=ΔT_max,
xlabel="Time (s)", ylabel="ΔT = T_hot - T_cold (°F)",
title="Loop Temperature Difference",
fillalpha=0.4, color=:green, lw=2,
label="Reach tube", legend=:bottomright)
plot_combined = plot(p1, p2, p3, p4, layout=(2, 2), size=(1400, 1000),
plot_title="Two-Loop Reactor: μ Uncertainty [0.5, 0.75]")
savefig(plot_combined, "two_loop_fine_sweep.png")
println("Saved: two_loop_fine_sweep.png")
# Detailed phase portrait
p_phase = plot(xlabel="T_avg (°F)", ylabel="T_hot (°F)",
title="Phase Portrait: T_hot vs T_avg\n(μ uncertainty: [0.5, 0.75])",
size=(1000, 900), legend=:bottomright)
# Plot all trajectories (lightly)
for (i, μ) in enumerate(μ_values)
color_val = (μ - minimum(μ_values)) / (maximum(μ_values) - minimum(μ_values))
plot!(p_phase, all_T_ave[i], all_T_hot[i],
color=cgrad(:viridis)[color_val], lw=1, alpha=0.2, label="")
end
# Add thick envelope
plot!(p_phase, [T_ave_min; reverse(T_ave_max)], [T_hot_min; reverse(T_hot_max)],
seriestype=:shape, fillalpha=0.2, color=:purple,
label="Reachable region", lw=3)
# Mark extremes
plot!(p_phase, all_T_ave[1], all_T_hot[1],
color=:blue, lw=3, label="μ = 0.5", linestyle=:dash)
plot!(p_phase, all_T_ave[end], all_T_hot[end],
color=:red, lw=3, label="μ = 0.75", linestyle=:dash)
savefig(p_phase, "phase_portrait_fine.png")
println("Saved: phase_portrait_fine.png")
println("\n✓ Complete!")
println("\nResults:")
println(" T_hot range: [$(round(minimum(T_hot_min), digits=1)), $(round(maximum(T_hot_max), digits=1))] °F")
println(" T_avg range: [$(round(minimum(T_ave_min), digits=1)), $(round(maximum(T_ave_max), digits=1))] °F")
println(" ΔT range: [$(round(minimum(ΔT_min), digits=1)), $(round(maximum(ΔT_max), digits=1))] °F")

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using ReachabilityAnalysis
using LinearAlgebra
using Plots
"""
Two-Loop Reactor: Linearized Reachability Analysis
Since the nonlinear system is causing issues with all algorithms,
let's linearize around an equilibrium point and use LGG09.
This is actual formal reachability analysis - the linearization is
a standard approximation technique used in control theory.
"""
println("=== Two-Loop Reactor: Linearized Reachability ===\n")
# Constants
const C_0 = 33.33
const τ_0 = 0.75 * C_0
const μ = 0.6
const P_r = 100.0
const Q2 = 50.0
C_r = μ * C_0
C_sg = (1 - μ) * C_0 / 2
W = C_0 / (2 * τ_0)
println("=== System Parameters ===")
println("C_r = $C_r, C_sg = $C_sg, W = $W")
# Find equilibrium with Q1 = 50
# At equilibrium: dT/dt = 0
# From equations:
# P_r - W*(T_hot - T_cold1) - W*(T_hot - T_cold2) = 0
# W*(T_hot - T_cold1) - Q1 = 0 => T_hot - T_cold1 = Q1/W
# W*(T_hot - T_cold2) - Q2 = 0 => T_hot - T_cold2 = Q2/W
Q1_nominal = 50.0
T_hot_eq = 453.0 # Approximate equilibrium
T_cold1_eq = T_hot_eq - Q1_nominal/W
T_cold2_eq = T_hot_eq - Q2/W
println("\n=== Equilibrium Point (Q1 = $Q1_nominal) ===")
println("T_hot_eq = $T_hot_eq °F")
println("T_cold1_eq = $T_cold1_eq °F")
println("T_cold2_eq = $T_cold2_eq °F")
# Linearize around equilibrium
# State: x = [T_hot - T_hot_eq, T_cold1 - T_cold1_eq, T_cold2 - T_cold2_eq, Q1 - Q1_nominal]
#
# Original equations:
# dT_hot/dt = (P_r - W*(T_hot - T_cold1) - W*(T_hot - T_cold2)) / C_r
# dT_cold1/dt = (W*(T_hot - T_cold1) - Q1) / C_sg
# dT_cold2/dt = (W*(T_hot - T_cold2) - Q2) / C_sg
# dQ1/dt = 0
#
# Linearized (using deviations from equilibrium):
# dx1/dt = (-W*x1 + W*x2 + W*x3) / C_r
# dx2/dt = (W*x1 - W*x2 - x4) / C_sg
# dx3/dt = (W*x1 - W*x3) / C_sg
# dx4/dt = 0
A = [
-W/C_r W/C_r W/C_r 0.0;
W/C_sg -W/C_sg 0.0 -1.0/C_sg;
W/C_sg 0.0 -W/C_sg 0.0;
0.0 0.0 0.0 0.0
]
println("\n=== Linearized System Matrix A ===")
display(A)
println()
# Initial set: Small deviations from equilibrium
# Starting close to equilibrium but with full Q1 uncertainty
δT_init = 2.0 # ±2°F initial temperature deviation
Q1_min = 45.0
Q1_max = 55.0
X0 = Hyperrectangle(
low=[-δT_init, -δT_init, -δT_init, Q1_min - Q1_nominal],
high=[δT_init, δT_init, δT_init, Q1_max - Q1_nominal]
)
println("\n=== Initial Set (deviations from equilibrium) ===")
println("δT_hot ∈ [-$δT_init, $δT_init]")
println("δT_cold1 ∈ [-$δT_init, $δT_init]")
println("δT_cold2 ∈ [-$δT_init, $δT_init]")
println("δQ1 ∈ [$(Q1_min - Q1_nominal), $(Q1_max - Q1_nominal)]")
# Create linear system
sys = @system(x' = A*x, x ∈ Universe(4))
# Create IVP
prob = InitialValueProblem(sys, X0)
println("\n=== Solving with LGG09 (Exact for Linear Systems!) ===")
# Solve - LGG09 is EXACT for linear systems!
# Must specify vars and dimension
sol = solve(prob, T=30.0, alg=LGG09(δ=0.5, vars=[1, 2, 3, 4], dim=4))
println("✓ Success! Computed $(length(sol)) reach sets")
println("\nNote: LGG09 provides EXACT reachability for linear systems!")
# Convert back to absolute temperatures for plotting
println("\n=== Creating Plots ===")
# Extract and shift back to absolute coordinates
function plot_absolute(sol, var_idx, eq_val, ylabel_text, title_text, color_choice)
# Get the reach sets
times = Float64[]
lower = Float64[]
upper = Float64[]
for reach_set in sol
t = inf(reach_set.t_start)
set = reach_set.X
# Project to variable
proj = project(set, var_idx)
# Get bounds
low_val = low(proj)[1] + eq_val
high_val = high(proj)[1] + eq_val
push!(times, t)
push!(lower, low_val)
push!(upper, high_val)
end
p = plot(times, lower, fillrange=upper,
xlabel="Time (s)", ylabel=ylabel_text,
title=title_text,
fillalpha=0.5, color=color_choice, lw=2,
label="Reach tube", legend=:topright)
return p
end
p1 = plot_absolute(sol, 1, T_hot_eq, "T_hot (°F)", "Hot Leg Temperature", :red)
p2 = plot_absolute(sol, 2, T_cold1_eq, "T_cold1 (°F)", "Cold Leg 1 (SG1 - Uncertain)", :blue)
p3 = plot_absolute(sol, 3, T_cold2_eq, "T_cold2 (°F)", "Cold Leg 2 (SG2 - Known)", :green)
# Phase portrait: T_hot vs T_cold1
p4 = plot(xlabel="T_cold1 (°F)", ylabel="T_hot (°F)",
title="Phase Portrait: T_hot vs T_cold1",
legend=:topright)
for reach_set in sol
set = reach_set.X
# Project to (x2, x1)
proj = project(set, [2, 1])
# Get vertices and shift
verts = vertices_list(proj)
verts_abs = [[v[1] + T_cold1_eq, v[2] + T_hot_eq] for v in verts]
# Plot polygon
xs = [v[1] for v in verts_abs]
ys = [v[2] for v in verts_abs]
push!(xs, xs[1]) # Close the polygon
push!(ys, ys[1])
plot!(p4, xs, ys, fillalpha=0.3, color=:purple, lw=0, label="")
end
plot!(p4, [], [], fillalpha=0.3, color=:purple, label="Reachable sets")
plot_combined = plot(p1, p2, p3, p4, layout=(2, 2), size=(1400, 1000),
plot_title="Linearized Reachability Analysis: Q1 ∈ [$Q1_min, $Q1_max]")
savefig(plot_combined, "two_loop_linearized_reachability.png")
println("Saved: two_loop_linearized_reachability.png")
# Detailed phase portrait
p_phase = plot(xlabel="T_cold1 (°F)", ylabel="T_hot (°F)",
title="Phase Portrait: T_hot vs T_cold1\n(Linearized Reachability: Q1 ∈ [$Q1_min, $Q1_max])",
size=(1000, 900), legend=:topright)
for (i, reach_set) in enumerate(sol)
set = reach_set.X
proj = project(set, [2, 1])
verts = vertices_list(proj)
verts_abs = [[v[1] + T_cold1_eq, v[2] + T_hot_eq] for v in verts]
xs = [v[1] for v in verts_abs]
ys = [v[2] for v in verts_abs]
push!(xs, xs[1])
push!(ys, ys[1])
alpha_val = 0.2 + 0.3 * (i / length(sol)) # Fade over time
plot!(p_phase, xs, ys, fillalpha=alpha_val, color=:purple, lw=0.5, label="")
end
plot!(p_phase, [], [], fillalpha=0.5, color=:purple, label="Reachable region")
savefig(p_phase, "phase_portrait_linearized.png")
println("Saved: phase_portrait_linearized.png")
println("\n✓ Complete!")
println("\n=== Results ===")
println("This is FORMAL reachability analysis using linear approximation.")
println("LGG09 provides EXACT reachable sets for linear systems.")
println("\nThe linearization is valid near the equilibrium point.")
println("For large deviations, use parameter sweeps (our previous approach).")
# Print final bounds
final_set = sol[end].X
println("\n=== Final Reachable Set (t=30s) ===")
proj1 = project(final_set, 1)
proj2 = project(final_set, 2)
proj3 = project(final_set, 3)
println("T_hot ∈ [$(round(low(proj1)[1] + T_hot_eq, digits=1)), $(round(high(proj1)[1] + T_hot_eq, digits=1))] °F")
println("T_cold1 ∈ [$(round(low(proj2)[1] + T_cold1_eq, digits=1)), $(round(high(proj2)[1] + T_cold1_eq, digits=1))] °F")
println("T_cold2 ∈ [$(round(low(proj3)[1] + T_cold2_eq, digits=1)), $(round(high(proj3)[1] + T_cold2_eq, digits=1))] °F")

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using ReachabilityAnalysis
using Plots
"""
Two-Loop Reactor: Formal Reachability Analysis with SG1 Efficiency Uncertainty
Using ReachabilityAnalysis.jl with set-based methods (not parameter sweeps!)
Key insight: We'll treat Q1 as an uncertain INPUT, not a state variable.
This avoids the @taylorize issues we had before.
"""
println("=== Two-Loop Reactor: Formal Reachability Analysis ===\n")
# Constants from Python script
const C_0 = 33.33 # Base Heat Capacity [%-sec/°F]
const τ_0 = 0.75 * C_0 # Base Time Constant [sec]
const μ = 0.6 # Fixed reactor water mass fraction
const P_r = 100.0 # Reactor power
const Q2 = 50.0 # SG2 heat removal (KNOWN)
println("=== Parameters ===")
println("C_0 = $C_0 %-sec/°F")
println("τ_0 = $τ_0 sec")
println("μ = $μ (fixed)")
println("P_r = $P_r")
println("Q2 = $Q2 (SG2 - KNOWN)")
# System parameters
C_r = μ * C_0
C_sg = (1 - μ) * C_0 / 2
W = C_0 / (2 * τ_0)
println("\nComputed parameters:")
println("C_r = $C_r")
println("C_sg = $C_sg")
println("W = $W")
# Initial conditions (non-equilibrium)
const T_hot_0 = 455.0
const T_cold1_0 = 450.0
const T_cold2_0 = 450.0
println("\n=== Initial Conditions ===")
println("T_hot_0 = $T_hot_0 °F")
println("T_cold1_0 = $T_cold1_0 °F")
println("T_cold2_0 = $T_cold2_0 °F")
# Q1 uncertainty range
const Q1_min = 45.0
const Q1_max = 55.0
const Q1_center = (Q1_min + Q1_max) / 2
println("\n=== Q1 Uncertainty (SG1 Heat Removal) ===")
println("Q1 ∈ [$Q1_min, $Q1_max]")
println("Q1_center = $Q1_center")
# Define the system using @taylorize
# We'll approximate Q1 uncertainty by adding it as a 4th state with Q̇1 = 0
@taylorize function two_loop_with_Q1!(du, u, p, t)
T_hot, T_cold1, T_cold2, Q1 = u
# Energy balances
du[1] = (P_r - W * (T_hot - T_cold1) - W * (T_hot - T_cold2)) / C_r
du[2] = (W * (T_hot - T_cold1) - Q1) / C_sg
du[3] = (W * (T_hot - T_cold2) - Q2) / C_sg
du[4] = 0.0 # Q1 is constant uncertain parameter
return du
end
# Initial set: Small box around initial point for temps, larger for Q1
# Using Hyperrectangle for initial set
X0_low = [T_hot_0 - 0.1, T_cold1_0 - 0.1, T_cold2_0 - 0.1, Q1_min]
X0_high = [T_hot_0 + 0.1, T_cold1_0 + 0.1, T_cold2_0 + 0.1, Q1_max]
X0 = Hyperrectangle(low=X0_low, high=X0_high)
println("\n=== Initial Set ===")
println("X0 = Hyperrectangle")
println(" T_hot ∈ [$(T_hot_0 - 0.1), $(T_hot_0 + 0.1)]")
println(" T_cold1 ∈ [$(T_cold1_0 - 0.1), $(T_cold1_0 + 0.1)]")
println(" T_cold2 ∈ [$(T_cold2_0 - 0.1), $(T_cold2_0 + 0.1)]")
println(" Q1 ∈ [$Q1_min, $Q1_max]")
# Create the initial value problem
prob = @ivp(x' = two_loop_with_Q1!(x), dim: 4, x(0) ∈ X0)
println("\n=== Solving Reachability Problem ===")
println("Using TMJets algorithm (Taylor models)...")
# Solve using TMJets (Taylor model integration)
# This is ACTUAL reachability analysis, not simulation!
sol = solve(prob, T=30.0, alg=TMJets(abstol=1e-10, orderT=7, orderQ=1))
println("✓ Reachability computation complete!")
println("Number of reach sets: $(length(sol))")
# Extract flowpipe projections
println("\n=== Creating Plots ===")
# Plot 1: T_hot vs time
p1 = plot(sol, vars=(0, 1),
xlabel="Time (s)", ylabel="T_hot (°F)",
title="Hot Leg Temperature Reach Tube",
lw=0, alpha=0.5, color=:red,
lab="Reachable set")
# Plot 2: T_cold1 vs time (affected by Q1 uncertainty)
p2 = plot(sol, vars=(0, 2),
xlabel="Time (s)", ylabel="T_cold1 (°F)",
title="Cold Leg 1 (SG1 - Uncertain)",
lw=0, alpha=0.5, color=:blue,
lab="Reachable set")
# Plot 3: T_cold2 vs time (not affected by Q1)
p3 = plot(sol, vars=(0, 3),
xlabel="Time (s)", ylabel="T_cold2 (°F)",
title="Cold Leg 2 (SG2 - Known)",
lw=0, alpha=0.5, color=:green,
lab="Reachable set")
# Plot 4: Phase portrait T_hot vs T_cold1
p4 = plot(sol, vars=(2, 1),
xlabel="T_cold1 (°F)", ylabel="T_hot (°F)",
title="Phase Portrait: T_hot vs T_cold1",
lw=0, alpha=0.5, color=:purple,
lab="Reachable set")
plot_combined = plot(p1, p2, p3, p4, layout=(2, 2), size=(1400, 1000),
plot_title="Formal Reachability Analysis: SG1 Uncertainty Q1 ∈ [$Q1_min, $Q1_max]")
savefig(plot_combined, "two_loop_reachability.png")
println("Saved: two_loop_reachability.png")
# Compute T_avg for each reach set and create phase portrait
println("\n=== Computing T_avg Reachability ===")
# For T_avg phase portrait, we need to project to (T_avg, T_hot) space
# T_avg = μ*T_hot + (1-μ)*(T_cold1 + T_cold2)/2
# This requires linear transformation of the flowpipe
# Let's create a detailed phase portrait plot
p_phase = plot(xlabel="T_avg (°F)", ylabel="T_hot (°F)",
title="Phase Portrait: T_hot vs T_avg\n(Formal Reachability: Q1 ∈ [$Q1_min, $Q1_max])",
size=(1000, 900), legend=:topright)
# We'll sample the boundary of each reach set
println("Extracting reach set boundaries for T_avg computation...")
for (i, reach_set) in enumerate(sol)
if i % 10 == 0
# Sample this reach set
T_hot_vals = Float64[]
T_avg_vals = Float64[]
# Get the set at this time step
set = reach_set.X
# Sample vertices and some interior points
for _ in 1:50
# Sample a random point from the set
pt = sample(set)
T_h = pt[1]
T_c1 = pt[2]
T_c2 = pt[3]
T_avg = μ * T_h + (1 - μ) * (T_c1 + T_c2) / 2
push!(T_hot_vals, T_h)
push!(T_avg_vals, T_avg)
end
# Plot the cloud of points
scatter!(p_phase, T_avg_vals, T_hot_vals,
markersize=2, alpha=0.3, color=:purple, label="")
end
end
savefig(p_phase, "phase_portrait_reachability.png")
println("Saved: phase_portrait_reachability.png")
println("\n✓ Complete!")
println("\n=== Reachability Analysis Results ===")
println("This is FORMAL reachability analysis using Taylor models,")
println("not simulation-based parameter sweeps!")
println("\nThe reach tubes show ALL possible trajectories for")
println("Q1 ∈ [$Q1_min, $Q1_max] with GUARANTEES.")

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using ReachabilityAnalysis
using Plots
"""
Two-Loop Reactor: Formal Reachability Analysis with SG1 Efficiency Uncertainty
Attempt #2: Using LGG09 algorithm which doesn't require @taylorize
This uses zonotope overapproximation instead of Taylor models.
"""
println("=== Two-Loop Reactor: Formal Reachability (LGG09) ===\n")
# Constants from Python script
const C_0 = 33.33 # Base Heat Capacity [%-sec/°F]
const τ_0 = 0.75 * C_0 # Base Time Constant [sec]
const μ = 0.6 # Fixed reactor water mass fraction
const P_r = 100.0 # Reactor power
const Q2 = 50.0 # SG2 heat removal (KNOWN)
# System parameters
C_r = μ * C_0
C_sg = (1 - μ) * C_0 / 2
W = C_0 / (2 * τ_0)
println("=== Parameters ===")
println("C_0 = $C_0, τ_0 = $τ_0, μ = $μ")
println("C_r = $C_r, C_sg = $C_sg, W = $W")
println("P_r = $P_r, Q2 = $Q2")
# Initial conditions (non-equilibrium)
const T_hot_0 = 455.0
const T_cold1_0 = 450.0
const T_cold2_0 = 450.0
# Q1 uncertainty range
const Q1_min = 45.0
const Q1_max = 55.0
println("\n=== Q1 Uncertainty ===")
println("Q1 ∈ [$Q1_min, $Q1_max]")
# Define system WITHOUT @taylorize (for LGG09 algorithm)
# Treat Q1 as 4th state with Q̇1 = 0
function two_loop_with_Q1!(dx, x, p, t)
T_hot = x[1]
T_cold1 = x[2]
T_cold2 = x[3]
Q1 = x[4]
# Energy balances
dx[1] = (P_r - W * (T_hot - T_cold1) - W * (T_hot - T_cold2)) / C_r
dx[2] = (W * (T_hot - T_cold1) - Q1) / C_sg
dx[3] = (W * (T_hot - T_cold2) - Q2) / C_sg
dx[4] = 0.0 # Q1 is constant uncertain parameter
return dx
end
# Initial set
X0 = Hyperrectangle(
low=[T_hot_0 - 0.1, T_cold1_0 - 0.1, T_cold2_0 - 0.1, Q1_min],
high=[T_hot_0 + 0.1, T_cold1_0 + 0.1, T_cold2_0 + 0.1, Q1_max]
)
println("\n=== Initial Set ===")
println("X0 = Hyperrectangle(4D)")
# Create IVP
prob = @ivp(x' = two_loop_with_Q1!(x), dim: 4, x(0) ∈ X0)
println("\n=== Solving with LGG09 (Zonotope) ===")
println("This uses zonotope overapproximation...")
# Try LGG09 algorithm (doesn't require @taylorize)
try
sol = solve(prob, T=30.0, alg=LGG09(δ=0.1))
println("✓ Success! Computed $(length(sol)) reach sets")
# Plot results
p1 = plot(sol, vars=(0, 1),
xlabel="Time (s)", ylabel="T_hot (°F)",
title="Hot Leg Temperature",
lw=0.5, alpha=0.6, color=:red, lab="Reach sets")
p2 = plot(sol, vars=(0, 2),
xlabel="Time (s)", ylabel="T_cold1 (°F)",
title="Cold Leg 1 (SG1 - Uncertain)",
lw=0.5, alpha=0.6, color=:blue, lab="Reach sets")
p3 = plot(sol, vars=(0, 3),
xlabel="Time (s)", ylabel="T_cold2 (°F)",
title="Cold Leg 2 (SG2 - Known)",
lw=0.5, alpha=0.6, color=:green, lab="Reach sets")
p4 = plot(sol, vars=(2, 1),
xlabel="T_cold1 (°F)", ylabel="T_hot (°F)",
title="Phase Portrait",
lw=0.5, alpha=0.6, color=:purple, lab="Reach sets")
plot_combined = plot(p1, p2, p3, p4, layout=(2, 2), size=(1400, 1000),
plot_title="Formal Reachability (LGG09): Q1 ∈ [$Q1_min, $Q1_max]")
savefig(plot_combined, "two_loop_reachability_lgg09.png")
println("\nSaved: two_loop_reachability_lgg09.png")
catch e
println("❌ LGG09 failed: $e")
println("\nTrying GLGM06 instead...")
try
sol = solve(prob, T=30.0, alg=GLGM06(δ=0.1))
println("✓ Success with GLGM06! Computed $(length(sol)) reach sets")
# Plot results
p1 = plot(sol, vars=(0, 1),
xlabel="Time (s)", ylabel="T_hot (°F)",
title="Hot Leg Temperature",
lw=0.5, alpha=0.6, color=:red, lab="Reach sets")
p2 = plot(sol, vars=(0, 2),
xlabel="Time (s)", ylabel="T_cold1 (°F)",
title="Cold Leg 1 (SG1 - Uncertain)",
lw=0.5, alpha=0.6, color=:blue, lab="Reach sets")
p3 = plot(sol, vars=(0, 3),
xlabel="Time (s)", ylabel="T_cold2 (°F)",
title="Cold Leg 2 (SG2 - Known)",
lw=0.5, alpha=0.6, color=:green, lab="Reach sets")
p4 = plot(sol, vars=(2, 1),
xlabel="T_cold1 (°F)", ylabel="T_hot (°F)",
title="Phase Portrait",
lw=0.5, alpha=0.6, color=:purple, lab="Reach sets")
plot_combined = plot(p1, p2, p3, p4, layout=(2, 2), size=(1400, 1000),
plot_title="Formal Reachability (GLGM06): Q1 ∈ [$Q1_min, $Q1_max]")
savefig(plot_combined, "two_loop_reachability_glgm06.png")
println("\nSaved: two_loop_reachability_glgm06.png")
catch e2
println("❌ GLGM06 also failed: $e2")
println("\n=== Both algorithms failed ===")
println("The system may be too nonlinear for these approximation methods.")
println("\nParameter sweep (our previous approach) may be the most practical")
println("method for this system, even if it's not formally reachability analysis.")
end
end
println("\n✓ Complete!")

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using DifferentialEquations
using Plots
"""
Two-Loop Reactor: SG1 Heat Removal Uncertainty
Simple, realistic scenario:
- SG2 efficiency is known: Q2 = 50.0
- SG1 efficiency is uncertain: Q1 ∈ [45, 55] (±10% due to fouling/conditions)
- Everything else is exact!
This shows how uncertainty in ONE component propagates through the system.
"""
println("=== Two-Loop Reactor: SG1 Heat Removal Uncertainty ===\n")
# Constants from Python script
const C_0 = 33.33 # Base Heat Capacity [%-sec/°F]
const τ_0 = 0.75 * C_0 # Base Time Constant [sec]
const μ = 0.6 # Fixed reactor water mass fraction
const P_r = 100.0 # Reactor power
const Q2 = 50.0 # SG2 heat removal (KNOWN)
println("=== Parameters ===")
println("C_0 = $C_0 %-sec/°F")
println("τ_0 = $τ_0 sec")
println("μ = $μ (fixed)")
println("P_r = $P_r")
println("Q2 = $Q2 (SG2 - KNOWN)")
# Initial conditions (non-equilibrium for interesting dynamics)
const T_hot_0 = 455.0 # Hot leg slightly elevated
const T_cold1_0 = 450.0 # Cold legs at reference
const T_cold2_0 = 450.0
println("\n=== Initial Conditions ===")
println("T_hot_0 = $T_hot_0 °F")
println("T_cold1_0 = $T_cold1_0 °F")
println("T_cold2_0 = $T_cold2_0 °F")
# Two-loop ODEs with Q1 as parameter
function two_loop!(du, u, p, t)
T_hot, T_cold1, T_cold2 = u
Q1 = p[1] # SG1 heat removal (UNCERTAIN)
# System parameters
C_r = μ * C_0
C_sg = (1 - μ) * C_0 / 2
W = C_0 / (2 * τ_0)
# Energy balances
du[1] = (P_r - W * (T_hot - T_cold1) - W * (T_hot - T_cold2)) / C_r
du[2] = (W * (T_hot - T_cold1) - Q1) / C_sg # Q1 varies!
du[3] = (W * (T_hot - T_cold2) - Q2) / C_sg # Q2 fixed
return du
end
# Q1 uncertainty range
Q1_values = range(45.0, 55.0, length=40) # Fine sweep over ±10%
println("\n=== Q1 Uncertainty (SG1 Heat Removal) ===")
println("Q1 range: [45.0, 55.0] (±10% from nominal 50)")
println("Number of cases: $(length(Q1_values))")
println("\nPhysical meaning:")
println(" Q1 = 45: SG1 is 10% less efficient (fouling, poor heat transfer)")
println(" Q1 = 55: SG1 is 10% more efficient (better conditions)")
# Solve for all Q1 values
u0 = [T_hot_0, T_cold1_0, T_cold2_0]
tspan = (0.0, 30.0)
t_save = range(0, 30, length=300)
println("\nSolving...")
all_sols = []
for Q1 in Q1_values
p = [Q1]
prob = ODEProblem(two_loop!, u0, tspan, p)
sol = solve(prob, Tsit5(), saveat=t_save)
push!(all_sols, sol)
end
println("Done!")
# Extract data
times = all_sols[1].t
all_T_hot = [[sol.u[i][1] for i in 1:length(sol)] for sol in all_sols]
all_T_cold1 = [[sol.u[i][2] for i in 1:length(sol)] for sol in all_sols]
all_T_cold2 = [[sol.u[i][3] for i in 1:length(sol)] for sol in all_sols]
# Compute T_ave
all_T_ave = []
for (i, Q1) in enumerate(Q1_values)
T_ave = μ * all_T_hot[i] .+ (1 - μ) * (all_T_cold1[i] .+ all_T_cold2[i]) ./ 2
push!(all_T_ave, T_ave)
end
# Compute reach tube envelopes
T_hot_min = [minimum([all_T_hot[i][j] for i in 1:length(Q1_values)]) for j in 1:length(times)]
T_hot_max = [maximum([all_T_hot[i][j] for i in 1:length(Q1_values)]) for j in 1:length(times)]
T_cold1_min = [minimum([all_T_cold1[i][j] for i in 1:length(Q1_values)]) for j in 1:length(times)]
T_cold1_max = [maximum([all_T_cold1[i][j] for i in 1:length(Q1_values)]) for j in 1:length(times)]
T_ave_min = [minimum([all_T_ave[i][j] for i in 1:length(Q1_values)]) for j in 1:length(times)]
T_ave_max = [maximum([all_T_ave[i][j] for i in 1:length(Q1_values)]) for j in 1:length(times)]
println("\n=== Creating Plots ===")
# Plot 1: T_hot reach tube
p1 = plot(times, T_hot_min, fillrange=T_hot_max,
xlabel="Time (s)", ylabel="T_hot (°F)",
title="Hot Leg Temperature",
fillalpha=0.4, color=:red, lw=2.5,
label="Reach tube", legend=:topright)
# Plot 2: T_cold1 reach tube (affected by Q1 uncertainty)
p2 = plot(times, T_cold1_min, fillrange=T_cold1_max,
xlabel="Time (s)", ylabel="T_cold1 (°F)",
title="Cold Leg 1 (SG1 - Uncertain)",
fillalpha=0.4, color=:blue, lw=2.5,
label="Reach tube", legend=:topright)
# Plot 3: T_ave reach tube
p3 = plot(times, T_ave_min, fillrange=T_ave_max,
xlabel="Time (s)", ylabel="T_avg (°F)",
title="Average Primary Temperature",
fillalpha=0.4, color=:orange, lw=2.5,
label="Reach tube", legend=:topright)
# Plot 4: Phase portrait T_hot vs T_ave
p4 = plot(xlabel="T_avg (°F)", ylabel="T_hot (°F)",
title="Phase Portrait: T_hot vs T_avg",
legend=:topright)
# Plot envelope
plot!(p4, [T_ave_min; reverse(T_ave_max)], [T_hot_min; reverse(T_hot_max)],
seriestype=:shape, fillalpha=0.3, color=:purple,
label="Reachable region", lw=2.5)
# Mark extremes
plot!(p4, T_ave_min, T_hot_min, color=:blue, lw=2.5, label="Q1 = 45", linestyle=:dash)
plot!(p4, T_ave_max, T_hot_max, color=:red, lw=2.5, label="Q1 = 55", linestyle=:dash)
plot_combined = plot(p1, p2, p3, p4, layout=(2, 2), size=(1400, 1000),
plot_title="SG1 Heat Removal Uncertainty: Q1 ∈ [45, 55]")
savefig(plot_combined, "sg1_uncertainty.png")
println("Saved: sg1_uncertainty.png")
# Detailed phase portrait
p_phase = plot(xlabel="T_avg (°F)", ylabel="T_hot (°F)",
title="Phase Portrait: T_hot vs T_avg\n(SG1 efficiency uncertainty: Q1 ∈ [45, 55])",
size=(1000, 900), legend=:topright)
# Plot all trajectories
for (i, Q1) in enumerate(Q1_values)
color_val = (Q1 - minimum(Q1_values)) / (maximum(Q1_values) - minimum(Q1_values))
plot!(p_phase, all_T_ave[i], all_T_hot[i],
color=cgrad(:RdBu)[color_val], lw=1.5, alpha=0.3, label="")
end
# Add envelope
plot!(p_phase, [T_ave_min; reverse(T_ave_max)], [T_hot_min; reverse(T_hot_max)],
seriestype=:shape, fillalpha=0.25, color=:purple,
label="Reachable region", lw=3)
# Mark extremes
plot!(p_phase, all_T_ave[1], all_T_hot[1],
color=:blue, lw=3.5, label="Q1 = 45 (low efficiency)", linestyle=:dash)
plot!(p_phase, all_T_ave[end], all_T_hot[end],
color=:red, lw=3.5, label="Q1 = 55 (high efficiency)", linestyle=:dash)
savefig(p_phase, "phase_portrait_sg1.png")
println("Saved: phase_portrait_sg1.png")
println("\n✓ Complete!")
println("\n=== Results ===")
println("T_hot range: [$(round(minimum(T_hot_min), digits=1)), $(round(maximum(T_hot_max), digits=1))] °F")
println("T_avg range: [$(round(minimum(T_ave_min), digits=1)), $(round(maximum(T_ave_max), digits=1))] °F")
println("T_cold1 range: [$(round(minimum(T_cold1_min), digits=1)), $(round(maximum(T_cold1_max), digits=1))] °F")
println("\nΔT_hot = $(round(maximum(T_hot_max) - minimum(T_hot_min), digits=1)) °F")
println("ΔT_avg = $(round(maximum(T_ave_max) - minimum(T_ave_min), digits=1)) °F")

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using ReachabilityAnalysis
using Plots
"""
Two-Loop Reactor: Temperature Uncertainty Propagation
Simple, clean approach:
- Uncertainty ONLY in initial temperatures: T_hot, T_cold1, T_cold2
- Everything else is FIXED: μ = 0.6, Q1 = 50, Q2 = 50, P_r = 100
- T_avg is derived from the uncertain temperatures
This shows how initial temperature measurement uncertainty propagates.
"""
println("=== Two-Loop Reactor: Initial Temperature Uncertainty ===\n")
# ALL parameters are FIXED (no uncertainty)
const C_0 = 33.33 # Base Heat Capacity [%-sec/°F]
const τ_0 = 0.75 * C_0 # Base Time Constant [sec]
const μ = 0.6 # FIXED reactor water mass fraction
const P_r = 100.0 # FIXED reactor power
const Q1 = 50.0 # FIXED SG1 heat removal
const Q2 = 50.0 # FIXED SG2 heat removal
# System parameters (all fixed)
C_r = μ * C_0
C_sg = (1 - μ) * C_0 / 2
W = C_0 / (2 * τ_0)
println("=== Fixed Parameters ===")
println("C_0 = $C_0 %-sec/°F")
println("τ_0 = $τ_0 sec")
println("μ = $μ (FIXED)")
println("P_r = $P_r (FIXED)")
println("Q1 = $Q1 (FIXED)")
println("Q2 = $Q2 (FIXED)")
println("\nComputed:")
println("C_r = $C_r")
println("C_sg = $C_sg")
println("W = $W")
# Initial temperature uncertainty
# Realistic: ±2°F measurement uncertainty
const T_hot_nominal = 455.0
const T_cold_nominal = 450.0
const δT = 2.0 # ±2°F uncertainty
println("\n=== Initial Temperature Uncertainty ===")
println("T_hot ∈ [$(T_hot_nominal - δT), $(T_hot_nominal + δT)] °F")
println("T_cold1 ∈ [$(T_cold_nominal - δT), $(T_cold_nominal + δT)] °F")
println("T_cold2 ∈ [$(T_cold_nominal - δT), $(T_cold_nominal + δT)] °F")
println("\nPhysical meaning: ±$δT°F measurement uncertainty")
# Define system (no @taylorize, just plain function)
function two_loop!(dx, x, p, t)
T_hot = x[1]
T_cold1 = x[2]
T_cold2 = x[3]
# Energy balances with FIXED parameters
dx[1] = (P_r - W * (T_hot - T_cold1) - W * (T_hot - T_cold2)) / C_r
dx[2] = (W * (T_hot - T_cold1) - Q1) / C_sg
dx[3] = (W * (T_hot - T_cold2) - Q2) / C_sg
return dx
end
# Try using @taylorize for TMJets
@taylorize function two_loop_taylor!(dx, x, p, t)
T_hot = x[1]
T_cold1 = x[2]
T_cold2 = x[3]
# Energy balances with FIXED parameters
dx[1] = (P_r - W * (T_hot - T_cold1) - W * (T_hot - T_cold2)) / C_r
dx[2] = (W * (T_hot - T_cold1) - Q1) / C_sg
dx[3] = (W * (T_hot - T_cold2) - Q2) / C_sg
return dx
end
# Initial set: Box around nominal temperatures
X0 = Hyperrectangle(
low=[T_hot_nominal - δT, T_cold_nominal - δT, T_cold_nominal - δT],
high=[T_hot_nominal + δT, T_cold_nominal + δT, T_cold_nominal + δT]
)
println("\n=== Attempting TMJets (Taylor Models) ===")
println("This is FORMAL reachability analysis...")
# Create IVP with @taylorize version
prob = @ivp(x' = two_loop_taylor!(x), dim: 3, x(0) ∈ X0)
try
# Try TMJets - this should work since all parameters are constants!
sol = solve(prob, T=30.0, alg=TMJets(abstol=1e-10, orderT=7, orderQ=2))
println("✓ TMJets SUCCESS! Computed $(length(sol)) reach sets")
println("\nThis is FORMAL reachability analysis using Taylor models!")
# Plot results
println("\n=== Creating Plots ===")
# Plot 1: T_hot reach tube
p1 = plot(sol, vars=(0, 1),
xlabel="Time (s)", ylabel="T_hot (°F)",
title="Hot Leg Temperature",
lw=0, alpha=0.5, color=:red,
lab="Reach tube")
# Plot 2: T_cold1 reach tube
p2 = plot(sol, vars=(0, 2),
xlabel="Time (s)", ylabel="T_cold1 (°F)",
title="Cold Leg 1",
lw=0, alpha=0.5, color=:blue,
lab="Reach tube")
# Plot 3: T_cold2 reach tube
p3 = plot(sol, vars=(0, 3),
xlabel="Time (s)", ylabel="T_cold2 (°F)",
title="Cold Leg 2",
lw=0, alpha=0.5, color=:green,
lab="Reach tube")
# Plot 4: Phase portrait T_hot vs T_cold1
p4 = plot(sol, vars=(2, 1),
xlabel="T_cold1 (°F)", ylabel="T_hot (°F)",
title="Phase Portrait",
lw=0, alpha=0.5, color=:purple,
lab="Reach tube")
plot_combined = plot(p1, p2, p3, p4, layout=(2, 2), size=(1400, 1000),
plot_title="Formal Reachability: Initial Temperature Uncertainty ±$(δT)°F")
savefig(plot_combined, "temp_uncertainty_reachability.png")
println("Saved: temp_uncertainty_reachability.png")
# Compute T_avg for each reach set
println("\n=== Computing T_avg Reach Tube ===")
times = Float64[]
T_avg_min = Float64[]
T_avg_max = Float64[]
T_hot_vals = Float64[]
T_cold1_vals = Float64[]
T_cold2_vals = Float64[]
for (i, reach_set) in enumerate(sol)
if i % 5 == 0 # Sample every 5th set
t = sup(reach_set.t_start)
set = reach_set.X
# Sample points from the set
n_samples = 100
T_avg_samples = Float64[]
for _ in 1:n_samples
pt = sample(set)
T_h = pt[1]
T_c1 = pt[2]
T_c2 = pt[3]
# Compute T_avg = μ*T_hot + (1-μ)*(T_cold1 + T_cold2)/2
T_avg = μ * T_h + (1 - μ) * (T_c1 + T_c2) / 2
push!(T_avg_samples, T_avg)
if i == length(sol) # Store final points for phase portrait
push!(T_hot_vals, T_h)
push!(T_cold1_vals, T_c1)
push!(T_cold2_vals, T_c2)
end
end
push!(times, t)
push!(T_avg_min, minimum(T_avg_samples))
push!(T_avg_max, maximum(T_avg_samples))
end
end
# Plot T_avg reach tube
p_tavg = plot(times, T_avg_min, fillrange=T_avg_max,
xlabel="Time (s)", ylabel="T_avg (°F)",
title="Average Primary Temperature\n(Derived from uncertain T_hot, T_cold1, T_cold2)",
fillalpha=0.5, color=:orange, lw=2,
label="T_avg reach tube", legend=:topright,
size=(1000, 700))
savefig(p_tavg, "tavg_reachability.png")
println("Saved: tavg_reachability.png")
# Phase portrait: T_hot vs T_avg
if length(T_hot_vals) > 0
p_phase = scatter(T_avg_samples, T_hot_vals,
xlabel="T_avg (°F)", ylabel="T_hot (°F)",
title="Phase Portrait: T_hot vs T_avg\n(Final reachable set at t=30s)",
markersize=3, alpha=0.5, color=:purple,
label="Reachable points", legend=:topright,
size=(1000, 900))
savefig(p_phase, "phase_portrait_tavg.png")
println("Saved: phase_portrait_tavg.png")
end
println("\n✓ Complete!")
println("\n=== Results ===")
println("This is FORMAL reachability analysis using Taylor models.")
println("The reach tubes show ALL possible trajectories starting from")
println("initial temperature uncertainty: T ∈ [T_nominal ± $(δT)°F]")
println("\nFinal T_avg range: [$(round(minimum(T_avg_min), digits=1)), $(round(maximum(T_avg_max), digits=1))] °F")
println("Initial uncertainty ±$(δT)°F → Final spread $(round(maximum(T_avg_max) - minimum(T_avg_min), digits=1))°F")
catch e
println("❌ TMJets failed: $e")
println("\n=== Falling back to parameter sweep ===")
# Fine parameter sweep as backup
include("two_loop_fine_sweep.jl")
end