journal: scaffold + 2 retroactive invention-log entries
journal/ directory, LaTeX-based, dated entries, callout boxes for
derivations / decisions / dead ends / limitations, plus an \apass{}
macro for in-line markers when a later deep-pass is needed.
Retroactive A-style entries for 2026-04-17 (controllers, linearization,
LQR, operation-mode linear reach, Lyapunov barrier) and 2026-04-20
(predicates restructure into deadbands+safety+invariants, OL-vs-CL
barrier analysis, mode-obligation taxonomy, heatup-rate-as-halfspace,
mode_boundaries, first Julia nonlinear reach attempt).
Both entries include derivations written out in math, dead-ends I
hit, code snippets with commentary, figure embeds, and terminal
output where it changed what we did next. The goal is invention-log
depth — readable 4 years from now without the git history to help.
journal/README.md documents the conventions. journal.tex aggregates
all entries into one PDF via latexmk.
Kept claude_memory/ separate as per earlier agreement — those are
short AI-context notes, different audience.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
This commit is contained in:
Dane Sabo
parent
a56fcbedc2
commit
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15
journal/.gitignore
vendored
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journal/.gitignore
vendored
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# LaTeX build artifacts
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*.aux
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*.bbl
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*.bcf
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*.blg
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*.fdb_latexmk
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*.fls
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*.log
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*.out
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*.run.xml
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*.synctex.gz
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*.toc
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*.lof
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*.lot
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*.pdf
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124
journal/README.md
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124
journal/README.md
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# Lab Journal
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|
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The HAHACS invention log for the `pwr-hybrid-3-demo` preliminary
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example. Each dated session gets an entry. The goal: a reader in 2030
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should be able to rebuild the thesis work from this journal alone.
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## Structure
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```
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journal/
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preamble.tex shared LaTeX setup (fonts, listings, callouts, macros)
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journal.tex top-level aggregator (builds all entries into one PDF)
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entries/ dated session entries, one file per session
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YYYY-MM-DD-{topic-slug}.tex
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figures/ journal-specific figures (referenced from entries)
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README.md this file
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```
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## Conventions
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### Filename
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`entries/YYYY-MM-DD-{topic-slug}.tex`. One file per session. If a day has multiple distinct sessions, use time-of-day in the slug (e.g., `2026-04-20-morning-predicates.tex` and `2026-04-20-evening-mega-session.tex`).
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### Entry skeleton
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```tex
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\session{2026-04-17}{duration}{one-line summary}
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\section{Session: ... (YYYY-MM-DD)}
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\label{sec:YYYYMMDD}
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\subsection*{Goal}
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What I set out to do and why.
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\subsection*{What landed}
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...
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\subsection*{Key decisions}
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...
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\subsection*{Dead ends}
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...
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\subsection*{Derivations}
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...
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\subsection*{Results}
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...
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\subsection*{Limitations recorded}
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...
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\subsection*{Open at close}
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...
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```
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Entries compile standalone (each starts with `\input{../preamble.tex}` wrapped in a conditional so it only pulls preamble when not already loaded by `journal.tex`), or together via `journal.tex`.
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### Two entry styles
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- **A-style (deep / invention-log)**: full derivations, code commentary, dead-ends, embedded figures, terminal output where useful. Used for retroactive entries and sessions that land meaningful artifacts.
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- **B-style (narrative + pointers)**: end-of-session notes. Uses `\apass{...}` callouts to flag spots that need a later A-pass.
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### Callout boxes
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From `preamble.tex`:
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| Environment | Use |
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|---|---|
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| `derivation` | Math derivations |
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| `decision` | Design choices with rationale + alternatives |
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| `deadend` | Paths that didn't work |
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| `limitation` | Soundness gaps, known-approximate behavior |
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Plus the inline `\apass{text}` marker for A-pass TODOs.
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### Code inclusion
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- `\juliafile[options]{path/to/file.jl}` — includes a Julia source file as a numbered listing.
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- `\matlabfile[...]{...}` — for MATLAB sources.
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- Or inline with `lstlisting` environment and `language=Julia` for snippets.
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Always include the path as the listing caption so readers can find the file.
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### Figures
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Figures live in `../docs/figures/` (shared with the thesis) or `figures/` (journal-only). The preamble sets `\graphicspath` to check both.
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Always include:
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1. A descriptive caption (what's on axes, what's being shown).
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2. A discussion in the surrounding prose — *what the figure proves or illustrates*. Figures without discussion are noise.
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### Terminal output
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For a numerical result or an error that drove a decision, include the actual terminal text in a `lstlisting` block with `style=terminal`:
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```tex
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\begin{lstlisting}[style=terminal]
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TMJets: 10583 reach-sets
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T_c envelope: [274.45, 295.0] C
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FAILED: AssertionError: radius must be nonnegative
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\end{lstlisting}
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```
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Don't include full logs — only the lines that changed what you did next.
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## Build
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```bash
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cd journal
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latexmk -pdf journal.tex # whole journal as one PDF
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latexmk -pdf entries/2026-04-17-controllers-linear-reach.tex # one entry
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```
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Requires TeX Live with `tcolorbox`, `listings`, `inconsolata`, `siunitx`, `cleveref`, `hyperref`, `fancyhdr`. All in the standard distribution.
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## Not a replacement for
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- `claude_memory/` — short AI-context notes, Markdown, different audience.
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- `reachability/WALKTHROUGH.md` — standalone doc summarizing current state of reach analysis.
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- Git commit messages — per-commit rationale for code changes.
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The journal is the *chronological narrative* of the work. The others are snapshots, summaries, or pointers. They're all legitimate; they do different things.
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681
journal/entries/2026-04-17-controllers-linear-reach.tex
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681
journal/entries/2026-04-17-controllers-linear-reach.tex
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% ---------------------------------------------------------------------------
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% 2026-04-17 — Controllers + linearization + operation-mode linear reach
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% Deep / A-style invention-log entry.
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% ---------------------------------------------------------------------------
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\session{2026-04-17}{full afternoon, $\sim$5 hr}{Build out the three
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missing DRC controllers (shutdown, heatup, scram), add a numerical
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linearization, design an LQR, and discharge the operation-mode safety
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obligation with a hand-rolled zonotope reach. Try a Lyapunov-ellipsoid
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barrier certificate; find it fundamentally too coarse for this plant.}
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\section{2026-04-17 — Controllers, linearization, operation-mode reach}
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\label{sec:20260417}
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\subsection*{Starting state}
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Coming into this session the repository had:
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\begin{itemize}
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\item A 10-state PKE + thermal-hydraulics plant model
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(\texttt{plant-model/pke\_th\_rhs.m}) and its parameters
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(\texttt{plant-model/pke\_params.m}).
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\item Two controllers: \texttt{ctrl\_null.m} (u=0 baseline) and
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\texttt{ctrl\_operation.m} (plain P on $T_{\mathrm{avg}}$).
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\item A FRET synthesis pipeline producing a discrete controller
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(DRC) with four modes (\texttt{q\_shutdown}, \texttt{q\_heatup},
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\texttt{q\_operation}, \texttt{q\_scram}) and seven transitions.
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\item An empty \texttt{reachability/} directory. The thesis calls
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for per-mode reach analysis; nothing has been done yet.
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\end{itemize}
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\subsection*{Goal}
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Move the reachability thrust from zero to a first working artifact.
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To do that, the continuous side needs all four modes' controllers
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built out (can't reach-analyze a mode with no controller), a
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linearization tool (for LQR design and for linear reach set
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propagation), and a choice of reach tool. Build all that, then
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discharge one mode's safety obligation.
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\subsection*{Plant-model recap (what we're controlling)}
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The plant is a lumped point-kinetic reactor coupled to a three-node
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thermal loop. State vector $x \in \mathbb{R}^{10}$:
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$$x = \bigl[\,n,\; C_1,\; C_2,\; C_3,\; C_4,\; C_5,\; C_6,\; T_f,\;
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T_c,\; T_{\mathrm{cold}}\,\bigr]^\top$$
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with $n$ normalized power, $C_i$ the six delayed-neutron precursor
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concentrations, $T_f$ fuel temperature, $T_c$ average coolant
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temperature, $T_{\mathrm{cold}}$ cold-leg coolant temperature.
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Hot-leg temperature is algebraic: $T_{\mathrm{hot}} = 2T_c - T_{\mathrm{cold}}$
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(linear coolant profile).
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\begin{derivation}
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The full RHS $\dot x = f(x, u, w)$ is:
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\begin{align*}
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\rho &= u + \alpha_f(T_f - T_{f0}) + \alpha_c(T_c - T_{c0}) \\
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\dot n &= \frac{\rho - \beta}{\Lambda} n + \sum_{i=1}^{6} \lambda_i C_i \\
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\dot C_i &= \frac{\beta_i}{\Lambda} n - \lambda_i C_i \quad (i = 1,\dots,6) \\
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\dot T_f &= \frac{P_0 n - hA(T_f - T_c)}{M_f c_f} \\
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\dot T_c &= \frac{hA(T_f - T_c) - 2 W c_c (T_c - T_{\mathrm{cold}})}{M_c c_c} \\
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\dot T_{\mathrm{cold}} &= \frac{W c_c (T_{\mathrm{hot}} - T_{\mathrm{cold}}) - Q_{\mathrm{sg}}(t)}{M_{\mathrm{sg}} c_c}
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\end{align*}
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$u$ is external rod-worth reactivity (the control input).
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$Q_{\mathrm{sg}}(t)$ is the steam-generator heat-removal disturbance.
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All other symbols are plant constants (see
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\texttt{plant-model/pke\_params.m} for sourcing; most are representative
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of a 2-loop Westinghouse-class PWR).
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\end{derivation}
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\subsection*{Part 1: The three new controllers}
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The DRC has four modes; we had only one mode's continuous controller
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written. Need to fill in shutdown, heatup, and scram.
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\subsubsection*{Shutdown and scram — trivial, picked for physical defensibility}
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Both modes are passive: the reactor is parked deeply subcritical, rods
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are fully in, no feedback control is applied. Each is a one-liner.
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\begin{lstlisting}[language=Matlab,caption={\texttt{plant-model/controllers/ctrl\_shutdown.m}}]
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u = -5 * plant.beta; % ~5 $ subcritical
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\end{lstlisting}
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\begin{lstlisting}[language=Matlab,caption={\texttt{plant-model/controllers/ctrl\_scram.m}}]
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u = -8 * plant.beta; % ~8 $, typical regulatory 8-10 $ rod worth
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\end{lstlisting}
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\begin{decision}
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Constants, not feedback laws. Rationale:
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\begin{itemize}
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\item Real shutdown/scram behavior is ``rods fully inserted, worth
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$\sim$5--10~\$, no further action.'' A feedback law would be
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un-physical.
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\item Constants are trivial to reach-analyze. Closed-loop matrix
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$A_{\mathrm{cl}} = A$ (controller adds nothing to dynamics).
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\item At cold-start conditions ($T = T_{\mathrm{cold0}} \approx 290\,^\circ\mathrm{C}$),
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the linearized feedback from $\alpha_f, \alpha_c$ is about
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$+2.79\,\$$ of ``wants-to-be-supercritical'' reactivity. So
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$u = -5\,\$$ gives total $\rho \approx -2.2\,\$$, comfortably
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subcritical with margin for uncertainty. $-8\,\$$ on scram
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gives $\sim$$-5\,\$$ total — conservative.
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\item We could tighten to $-3\,\$$ and still be subcritical at
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warm temperatures, but the thesis wants a safety margin
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robust to any plausible state.
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\end{itemize}
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\end{decision}
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\subsubsection*{Heatup — the interesting controller}
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Heatup is the one transition mode that does real control work: drive
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$T_{\mathrm{avg}}$ from hot-standby conditions up to operating
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conditions while keeping the reactor bounded. This one needs design.
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The structure I landed on:
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\begin{equation}
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u(t, x) = \mathrm{sat}\Bigl(u_{\mathrm{ff}}(x) + K_p\bigl(T_{\mathrm{ref}}(t) - T_c\bigr),\; u_{\min},\; u_{\max}\Bigr)
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\end{equation}
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with three ingredients:
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\textbf{Feedback linearization.} Cancel the intrinsic temperature
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feedback exactly:
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\begin{equation}
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u_{\mathrm{ff}}(x) = -\alpha_f(T_f - T_{f0}) - \alpha_c(T_c - T_{c0})
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\end{equation}
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so that after substitution into the $\rho$ equation,
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\begin{equation}
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\rho_{\mathrm{total}} = u_{\mathrm{ff}} + K_p(T_{\mathrm{ref}} - T_c) + \underbrace{\alpha_f(T_f - T_{f0}) + \alpha_c(T_c - T_{c0})}_{\text{cancels } u_{\mathrm{ff}}}
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= K_p \cdot e
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\end{equation}
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where $e = T_{\mathrm{ref}} - T_c$. The cancellation is exact only if
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the controller's $(\alpha_f, \alpha_c)$ match the plant's. This is the
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seat of the controller's robustness risk; see \cref{sub:alpha-drift}.
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\textbf{Ramped reference.}
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\begin{equation}
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T_{\mathrm{ref}}(t) = \min\bigl(T_{\mathrm{start}} + r \cdot t,\; T_{\mathrm{target}}\bigr)
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\end{equation}
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with $r$ = 28~\unit{\celsius/\hour} (\num{7.78e-3}~\unit{\celsius/\second}),
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the tech-spec heatup rate. Clamped at $T_{\mathrm{target}}$ once the
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ramp reaches operating conditions.
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\textbf{Saturation on $u$.} Cap $|u|$ at $0.5\beta$ on the positive
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side and $5\beta$ on the negative side. The upper cap keeps
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$\rho < \beta$ (the prompt-criticality line) under any plausible
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state, which bounds the neutron-kinetics excursion rate for later
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reach analysis.
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\begin{deadend}
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\textbf{First-pass heatup controller, before saturation was added.}
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The initial design was just feedback-linearization + P with no
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saturation, starting from an initial condition $n = 10^{-6}$ (decay-heat
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level) at $T_{\mathrm{cold0}} = 290\,^\circ\mathrm{C}$ everywhere.
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|
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What went wrong: at that cold IC, the feedback terms give about
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|
$+2.79\,\$$ of positive reactivity. With $u_{\mathrm{ff}}$ cancelling
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those, $u = K_p \cdot e$ with $K_p = \num{1e-4}$. Error grows as
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$T_{\mathrm{ref}}$ ramps away from $T_c$ (which is still cold because
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the reactor has no power to heat it yet). Reactivity builds slowly,
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but because $n$ is tiny, power is tiny; the system takes
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\textbf{about 20 minutes} to generate enough power to heat the coolant,
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during which $T_c$ stays at 290~\unit{\celsius} while $T_{\mathrm{ref}}$
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ramps to 297. Once power catches up, the large accumulated error
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|
produces a power spike (visible in the first-pass figure: $n$ peaks
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near $0.018 = 1.8\%$), and the thermal mass overshoots
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$T_{\mathrm{target}}$ by about 7~\unit{\celsius} before the controller
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|
pulls it back.
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|
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Why it doesn't actually match physics: real plant heatup doesn't
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start from $n \approx 0$. Operators take the reactor critical at
|
||||||
|
low power \emph{in shutdown mode}, by slowly withdrawing rods, then
|
||||||
|
the DRC transitions to heatup with $n \sim 0.1\,\%$ already
|
||||||
|
established. Two fixes landed:
|
||||||
|
\begin{enumerate}
|
||||||
|
\item Saturation on $u$: prevents the accumulated-error spike.
|
||||||
|
\item Start heatup sims from a ``critical-at-low-power'' IC
|
||||||
|
($n = 10^{-3}$). \texttt{main\_mode\_sweep.m} uses this IC.
|
||||||
|
\end{enumerate}
|
||||||
|
|
||||||
|
The feedback-linearization controller alone doesn't know any of this
|
||||||
|
— it just does what the math says. The fixes are a controller design
|
||||||
|
change (saturation) and an IC assumption. A fancier heatup controller
|
||||||
|
would also include ramp-rate feedforward, but we don't need it yet.
|
||||||
|
\end{deadend}
|
||||||
|
|
||||||
|
\begin{decision}
|
||||||
|
\textbf{No integrator.} We stay with P-only (no PI), even knowing
|
||||||
|
there will be structural ramp-tracking lag on the order of
|
||||||
|
$r/K_p/\omega_{\mathrm{thermal}}$. Rationale:
|
||||||
|
\begin{itemize}
|
||||||
|
\item The plant already has strong integrators via thermal mass
|
||||||
|
($M_c c_c$, $M_{\mathrm{sg}} c_c$). PI would double-integrate.
|
||||||
|
\item PI adds state, turning the 10-state reach problem into 11 states.
|
||||||
|
Modest cost by itself.
|
||||||
|
\item The dealbreaker: \textbf{anti-windup saturation makes PI a
|
||||||
|
hybrid system}. Each saturation region (unsat / low-sat /
|
||||||
|
high-sat) is its own continuous mode with its own dynamics.
|
||||||
|
Our outer hybrid automaton is the DRC; nesting an inner hybrid
|
||||||
|
system inside each DRC mode doubles the verification
|
||||||
|
complexity.
|
||||||
|
\item The DRC exits heatup on a predicate window
|
||||||
|
(\texttt{t\_avg\_in\_range} $\wedge$ \texttt{p\_above\_crit}),
|
||||||
|
not on zero steady-state error. So the structural lag is
|
||||||
|
acceptable.
|
||||||
|
\end{itemize}
|
||||||
|
\end{decision}
|
||||||
|
|
||||||
|
Final controller (after iteration) lives in
|
||||||
|
\texttt{plant-model/controllers/ctrl\_heatup.m}. The relevant body:
|
||||||
|
|
||||||
|
\begin{lstlisting}[language=Matlab,caption={\texttt{ctrl\_heatup.m} body (abbreviated)}]
|
||||||
|
Kp = 1e-4;
|
||||||
|
T_f = x(8); T_c = x(9); T_avg = T_c;
|
||||||
|
u_ff = -plant.alpha_f*(T_f - plant.T_f0) ...
|
||||||
|
-plant.alpha_c*(T_avg - plant.T_c0);
|
||||||
|
T_ref = min(ref.T_start + ref.ramp_rate * t, ref.T_target);
|
||||||
|
e = T_ref - T_avg;
|
||||||
|
u_unsat = u_ff + Kp * e;
|
||||||
|
u = min(max(u_unsat, -5*plant.beta), 0.5*plant.beta);
|
||||||
|
\end{lstlisting}
|
||||||
|
|
||||||
|
\subsection*{Part 2: Linearization}
|
||||||
|
|
||||||
|
We need $(A, B, B_w)$ matrices for (a) LQR gain synthesis and (b)
|
||||||
|
linear reach set propagation.
|
||||||
|
|
||||||
|
\begin{derivation}
|
||||||
|
Central finite differences on $f(x, u, w)$ at a chosen operating
|
||||||
|
point $(x^\star, u^\star, w^\star)$:
|
||||||
|
\begin{align*}
|
||||||
|
A_{:,k} &\approx \frac{f(x^\star + h_k e_k, u^\star, w^\star) - f(x^\star - h_k e_k, u^\star, w^\star)}{2 h_k} \\
|
||||||
|
B &\approx \frac{f(x^\star, u^\star + h_u, w^\star) - f(x^\star, u^\star - h_u, w^\star)}{2 h_u} \\
|
||||||
|
B_w &\approx \frac{f(x^\star, u^\star, w^\star + h_w) - f(x^\star, u^\star, w^\star - h_w)}{2 h_w}
|
||||||
|
\end{align*}
|
||||||
|
with $h_k = \max(\epsilon_{\mathrm{rel}} |x^\star_k|,\; \epsilon_{\mathrm{abs}})$
|
||||||
|
to handle the wildly different magnitudes in $x$ (e.g.\ $n \sim 1$
|
||||||
|
vs.\ $C_1 \sim 10^2$ vs.\ $T_f \sim 3 \times 10^2$).
|
||||||
|
\end{derivation}
|
||||||
|
|
||||||
|
Validation: simulate the linear model and the nonlinear model in
|
||||||
|
parallel under a small $Q_{\mathrm{sg}}$ step and compare deviations
|
||||||
|
from $x^\star$. \Cref{fig:linearize-sanity} shows the result.
|
||||||
|
|
||||||
|
\begin{figure}[h]
|
||||||
|
\centering
|
||||||
|
\includegraphics[width=0.85\linewidth]{linearize_sanity.png}
|
||||||
|
\caption{Linear vs.\ nonlinear sanity check under a 5\% $Q_{\mathrm{sg}}$
|
||||||
|
down-step at $t = 30$~\unit{\second}, $u = 0$. Linear (red dashed)
|
||||||
|
tracks nonlinear (blue solid) within $4 \times 10^{-4}$ relative
|
||||||
|
error at 60~\unit{\second}, improving to $5 \times 10^{-6}$ by
|
||||||
|
300~\unit{\second} as the system relaxes. The match is best on $n$
|
||||||
|
(tightly coupled), loosest on $C_3$ (slow precursor). Eigenvalues of
|
||||||
|
$A$ span $[-65.93, -0.0124]$~\unit{\per\second} — stiffness ratio
|
||||||
|
$\sim$5000. Conclusion: linearization is quantitatively trustworthy
|
||||||
|
for perturbations around $x^\star$ in a $\pm 50\,^\circ\mathrm{C}$
|
||||||
|
window.}
|
||||||
|
\label{fig:linearize-sanity}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
The code is a straightforward loop (see
|
||||||
|
\texttt{plant-model/pke\_linearize.m}); nothing subtle, but the
|
||||||
|
magnitude-aware step size matters — using a uniform $h = 10^{-6}$
|
||||||
|
either loses precision on the small states or truncates on the big
|
||||||
|
ones.
|
||||||
|
|
||||||
|
\subsection*{Part 3: LQR controller for operation mode}
|
||||||
|
|
||||||
|
With $(A, B)$ in hand, design a linear quadratic regulator around the
|
||||||
|
operating point.
|
||||||
|
|
||||||
|
\begin{derivation}
|
||||||
|
Solve the continuous-time algebraic Riccati equation (CARE) for
|
||||||
|
$X \succeq 0$:
|
||||||
|
$$A^\top X + X A - X B R^{-1} B^\top X + Q = 0$$
|
||||||
|
then set gain $K = R^{-1} B^\top X$. The closed-loop control law
|
||||||
|
$u = -K(x - x^\star)$ minimizes
|
||||||
|
$$\int_0^\infty \left[(x - x^\star)^\top Q (x - x^\star) + u^\top R u\right] dt$$
|
||||||
|
and guarantees $A_{\mathrm{cl}} = A - BK$ is Hurwitz whenever $(A,B)$
|
||||||
|
is stabilizable and $Q \succeq 0$.
|
||||||
|
\end{derivation}
|
||||||
|
|
||||||
|
Weight choices:
|
||||||
|
\begin{lstlisting}[language=Matlab,caption={LQR weights, from \texttt{ctrl\_operation\_lqr.m}}]
|
||||||
|
Q = diag([1, % n
|
||||||
|
1e-3, 1e-3, 1e-3, 1e-3, 1e-3, 1e-3, % precursors C_1..C_6
|
||||||
|
1e-2, % T_f
|
||||||
|
1e2, % T_c (the primary objective)
|
||||||
|
1]); % T_cold
|
||||||
|
R = 1e6;
|
||||||
|
\end{lstlisting}
|
||||||
|
|
||||||
|
\begin{decision}
|
||||||
|
$Q(9,9) = 10^2$ for $T_c$ is the primary design knob — heavy because
|
||||||
|
we care about the coolant average deviation. Precursor weights at
|
||||||
|
$10^{-3}$ because they're informational (not directly regulated).
|
||||||
|
$T_{\mathrm{cold}}$ at 1 because it's secondary but couples to $T_c$.
|
||||||
|
$R = 10^6$ balances so $|u|$ stays in the few-cent range — below prompt,
|
||||||
|
above sensor noise. Ballpark numbers; we can retune if the reach tube
|
||||||
|
comes out too tight or too loose.
|
||||||
|
\end{decision}
|
||||||
|
|
||||||
|
Head-to-head against plain P on the same 100\% $\to$ 80\%
|
||||||
|
$Q_{\mathrm{sg}}$ step (\cref{fig:p-vs-lqr}):
|
||||||
|
|
||||||
|
\begin{figure}[h]
|
||||||
|
\centering
|
||||||
|
\includegraphics[width=0.85\linewidth]{mode_sweep_op_P_vs_LQR.png}
|
||||||
|
\caption{P vs.\ LQR on operation mode under a 100\%$\to$80\%
|
||||||
|
$Q_{\mathrm{sg}}$ step at $t = 30$~\unit{\second}. Plain P
|
||||||
|
(\texttt{ctrl\_operation.m}) has a 5~\unit{\degreeFahrenheit} peak
|
||||||
|
overshoot and 0.8~\unit{\degreeFahrenheit} steady-state lag. LQR
|
||||||
|
(\texttt{ctrl\_operation\_lqr.m}) holds $T_{\mathrm{avg}}$
|
||||||
|
essentially at setpoint throughout. Zero extra state in either;
|
||||||
|
LQR is just using all ten state measurements instead of one.}
|
||||||
|
\label{fig:p-vs-lqr}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
\subsection*{Part 4: Running all modes end-to-end}
|
||||||
|
|
||||||
|
\texttt{plant-model/main\_mode\_sweep.m} exercises all five controllers
|
||||||
|
(null, shutdown, heatup, operation-P, operation-LQR, scram) back-to-back
|
||||||
|
from mode-appropriate ICs, and saves figures. The normalized-power
|
||||||
|
overview (\cref{fig:power-overview}) captures the whole sweep:
|
||||||
|
|
||||||
|
\begin{figure}[h]
|
||||||
|
\centering
|
||||||
|
\includegraphics[width=0.95\linewidth]{mode_sweep_power_overview.png}
|
||||||
|
\caption{Normalized power $n(t)$ across the four DRC modes
|
||||||
|
plus baseline. \emph{Shutdown} (top left, log scale): $n$ decays
|
||||||
|
from $10^{-6}$ to $10^{-12}$ over 10~\unit{\minute}. \emph{Heatup}
|
||||||
|
(top right): small power burst around minute 22 as the reactor
|
||||||
|
goes critical and heats. \emph{Operation} (bottom left): $Q_{\mathrm{sg}}$
|
||||||
|
step at 30~\unit{\second} drops $n$ to 78\% briefly before settling
|
||||||
|
at 80\%. \emph{Scram} (bottom right, log scale): $n$ drops $1 \to 10^{-6}$
|
||||||
|
via prompt jump + delayed-neutron tail over 10~\unit{\minute}. All
|
||||||
|
four behave as textbook would predict.}
|
||||||
|
\label{fig:power-overview}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
\subsection*{Part 5: Tool choice for reach analysis}
|
||||||
|
|
||||||
|
Three candidate tools were evaluated:
|
||||||
|
\begin{itemize}
|
||||||
|
\item \textbf{CORA} (MATLAB) — mature, stays in the existing
|
||||||
|
language, handles linear + nonlinear. Downside: $\sim$0.5~GB
|
||||||
|
install, heavy.
|
||||||
|
\item \textbf{JuliaReach} — newer, faster for large reach sets,
|
||||||
|
rigorous Taylor-model support. Downside: requires porting
|
||||||
|
the plant model to Julia.
|
||||||
|
\item \textbf{Flow* / SpaceEx} — C++ / no-longer-maintained,
|
||||||
|
both add toolchain friction.
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
\begin{decision}
|
||||||
|
For this first artifact: \textbf{write the reach tube by hand in pure
|
||||||
|
MATLAB}. Rationale: linear reach with bounded disturbance has a clean
|
||||||
|
analytic form — matrix exponential on state, augmented-matrix integral
|
||||||
|
for the disturbance contribution — that compiles to about 50 lines of
|
||||||
|
MATLAB. No toolbox install, no language switch. The result is a
|
||||||
|
sound box-shaped over-approximation.
|
||||||
|
|
||||||
|
If/when we need nonlinear reach, that's the point to stand up
|
||||||
|
JuliaReach. (Spoiler for the reader: this did happen in the
|
||||||
|
\texttt{2026-04-20-evening} session.)
|
||||||
|
\end{decision}
|
||||||
|
|
||||||
|
\subsection*{Part 6: The zonotope reach propagator}
|
||||||
|
|
||||||
|
For the LQR-closed-loop linear system
|
||||||
|
$\dot{\delta x} = A_{\mathrm{cl}} \delta x + B_w w$ with $\delta x(0) \in X_0$
|
||||||
|
(axis-aligned box) and $w \in [w_{\mathrm{lo}}, w_{\mathrm{hi}}]$, the
|
||||||
|
analytic solution splits into a homogeneous piece and a disturbance
|
||||||
|
integral:
|
||||||
|
|
||||||
|
\begin{derivation}
|
||||||
|
\begin{equation}
|
||||||
|
\delta x(t) = e^{A_{\mathrm{cl}} t} \delta x(0)
|
||||||
|
+ \int_0^t e^{A_{\mathrm{cl}}(t-s)} B_w w(s)\, ds.
|
||||||
|
\end{equation}
|
||||||
|
Discretize with step $\Delta t$ and let
|
||||||
|
$A_{\Delta} = e^{A_{\mathrm{cl}} \Delta t}$. The disturbance integral
|
||||||
|
over one step is
|
||||||
|
$$G_\Delta = \int_0^{\Delta t} e^{A_{\mathrm{cl}} s} B_w \, ds.$$
|
||||||
|
\textbf{Van Loan's trick (1978):} expand the matrix exponential of an
|
||||||
|
augmented matrix,
|
||||||
|
\begin{equation}
|
||||||
|
\exp\!\left(\begin{bmatrix} A & B_w \\ 0 & 0 \end{bmatrix} \Delta t\right)
|
||||||
|
= \begin{bmatrix} e^{A \Delta t} & G_\Delta \\ 0 & 1 \end{bmatrix}.
|
||||||
|
\end{equation}
|
||||||
|
Read the upper-right block — $G_\Delta$ is exact to machine precision
|
||||||
|
without numerical quadrature.
|
||||||
|
\end{derivation}
|
||||||
|
|
||||||
|
Now propagate a box $\delta x \in [c - h,\ c + h]$ (center $c$,
|
||||||
|
halfwidth $h$) through the linear map $M$:
|
||||||
|
|
||||||
|
\begin{derivation}
|
||||||
|
If $y = M \delta x$ and $\delta x_j \in [c_j - h_j,\ c_j + h_j]$, then
|
||||||
|
$$y_i = \sum_j M_{ij} \delta x_j \quad \Longrightarrow \quad y_i \in \bigl[\,M c \mp |M| h\,\bigr]_i.$$
|
||||||
|
The $\pm$ signs of $M_{ij}$ align with opposite corners of the
|
||||||
|
input box, so the sum extremum takes the absolute value of each
|
||||||
|
coefficient times the halfwidth. Conclusion: box propagation uses
|
||||||
|
\emph{elementwise} $|M|$, not signed $M$.
|
||||||
|
\end{derivation}
|
||||||
|
|
||||||
|
\begin{deadend}
|
||||||
|
\textbf{First version of \texttt{reach\_linear.m} used signed
|
||||||
|
$A_\Delta$ for halfwidth propagation.} Result: the reach tube came
|
||||||
|
out suspiciously tight — maximum $|T_c - T_{c0}|$ over 600~\unit{\second}
|
||||||
|
was exactly equal to the initial halfwidth, as if the disturbance
|
||||||
|
wasn't contributing at all.
|
||||||
|
|
||||||
|
Diagnosed after about 15~\unit{\minute} of confusion: when I wrote
|
||||||
|
$S_{\mathrm{half}} \leftarrow A_\Delta S_{\mathrm{half}} + G_\Delta w_{\mathrm{half}}$
|
||||||
|
with \emph{signed} $A_\Delta$, the positive and negative entries of
|
||||||
|
$A_\Delta$ across successive steps happened to cancel, so
|
||||||
|
$S_{\mathrm{half}}$ stayed near zero. The fix is to use elementwise
|
||||||
|
absolute values per the derivation above:
|
||||||
|
$$S_{\mathrm{half}} \leftarrow |A_\Delta| S_{\mathrm{half}} + |G_\Delta| w_{\mathrm{half}}.$$
|
||||||
|
Once the abs was added the disturbance halfwidth grew monotonically
|
||||||
|
as expected, and the reach tube gained a sensible envelope. This is
|
||||||
|
a case of ``the bug is easy to catch in the math, harder to catch
|
||||||
|
when you copy-pasted from the center-propagation code.''
|
||||||
|
\end{deadend}
|
||||||
|
|
||||||
|
The final propagator is the core of
|
||||||
|
\texttt{reachability/reach\_linear.m} (about 50~lines). The per-step
|
||||||
|
update is compact:
|
||||||
|
|
||||||
|
\begin{lstlisting}[language=Matlab,caption={Halfwidth-propagation core of \texttt{reach\_linear.m}}]
|
||||||
|
A_step = expm(A_cl * dt);
|
||||||
|
A_abs_step = abs(A_step);
|
||||||
|
M_aug = expm([A_cl, B_w; zeros(1, n+1)] * dt);
|
||||||
|
G_step = M_aug(1:n, n+1);
|
||||||
|
G_abs_step = abs(G_step);
|
||||||
|
|
||||||
|
for k = 2:M
|
||||||
|
x_center_now = A_step * x_center_now;
|
||||||
|
halfwidth_now = A_abs_step * halfwidth_now;
|
||||||
|
S_center = A_step * S_center + G_step * w_mid;
|
||||||
|
S_half = A_abs_step * S_half + G_abs_step * w_half;
|
||||||
|
|
||||||
|
center(:, k) = x_center_now + S_center;
|
||||||
|
R_lo(:, k) = center(:, k) - halfwidth_now - S_half;
|
||||||
|
R_hi(:, k) = center(:, k) + halfwidth_now + S_half;
|
||||||
|
end
|
||||||
|
\end{lstlisting}
|
||||||
|
|
||||||
|
\subsection*{Part 7: Operation-mode reach — discharging the safety obligation}
|
||||||
|
|
||||||
|
Entry box around $x_{\mathrm{op}}$:
|
||||||
|
\begin{itemize}
|
||||||
|
\item $n$: $\pm 1\%$ of nominal.
|
||||||
|
\item $C_i$: $\pm 0.1\%$ of equilibrium (precursors equilibrate
|
||||||
|
fast; tight entry).
|
||||||
|
\item $T_f,\ T_c,\ T_{\mathrm{cold}}$: $\pm 0.1$~\unit{\kelvin}.
|
||||||
|
\end{itemize}
|
||||||
|
Disturbance: $Q_{\mathrm{sg}} \in [0.85 P_0,\ 1.00 P_0]$ — a 15\%
|
||||||
|
load-down envelope (grid-following). Horizon: 600~\unit{\second}.
|
||||||
|
|
||||||
|
The reach envelope on $T_c$ is shown in \cref{fig:reach-tc}.
|
||||||
|
|
||||||
|
\begin{figure}[h]
|
||||||
|
\centering
|
||||||
|
\includegraphics[width=0.95\linewidth]{reach_operation_Tc.png}
|
||||||
|
\caption{Operation-mode reach tube on $T_{\mathrm{avg}}$, two views.
|
||||||
|
\emph{Left:} safety-band scale — reach tube (pink) is barely visible
|
||||||
|
because LQR holds it tight against the dashed $\pm 5$~\unit{\celsius}
|
||||||
|
safety band. \emph{Right:} zoomed to reveal the actual tube.
|
||||||
|
Halfwidth at $t=0$ is 0.1~\unit{\kelvin} (the entry box);
|
||||||
|
halfwidth at $t=600$~\unit{\second} is 0.033~\unit{\kelvin}. The
|
||||||
|
tube \emph{contracts} on the regulated direction — the signature
|
||||||
|
of tight feedback control. Max $|\delta T_c|$ over the horizon:
|
||||||
|
0.1~\unit{\kelvin} (the initial halfwidth dominates).}
|
||||||
|
\label{fig:reach-tc}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
Per-halfspace margins against \texttt{inv2\_holds} (the operation-mode
|
||||||
|
safety envelope):
|
||||||
|
|
||||||
|
\begin{lstlisting}[style=terminal]
|
||||||
|
=== Operation-mode reach vs inv2_holds safety limits ===
|
||||||
|
[fuel_centerline] a'x <= 1200.000 | max a'x = 328.934 | margin = +871.066 OK
|
||||||
|
[t_avg_high_trip] a'x <= 320.000 | max a'x = 308.449 | margin = +11.551 OK
|
||||||
|
[t_avg_low_trip] a'x <= -280.000 | max a'x = -308.249 | margin = +28.249 OK
|
||||||
|
[n_high_trip] a'x <= 1.150 | max a'x = 1.012 | margin = +0.138 OK
|
||||||
|
[n_low_operation] a'x <= -0.150 | max a'x = -0.842 | margin = +0.692 OK
|
||||||
|
[cold_leg_subcooled] a'x <= 305.000 | max a'x = 292.852 | margin = +12.148 OK
|
||||||
|
\end{lstlisting}
|
||||||
|
|
||||||
|
All six safety halfspaces pass. Tightest margin is
|
||||||
|
\texttt{n\_high\_trip} — LQR lets $n$ swing up to $\sim$1.01 to
|
||||||
|
compensate for load variation, leaving 12\% margin to the high-flux
|
||||||
|
trip. Temperatures have 10--870~\unit{\kelvin} margin each.
|
||||||
|
\textbf{Operation-mode obligation discharged}, subject to the
|
||||||
|
linearization caveat in \cref{sec:limitations-17}.
|
||||||
|
|
||||||
|
Per-state reach-set evolution (final halfwidth vs.\ initial):
|
||||||
|
|
||||||
|
\begin{center}
|
||||||
|
\small
|
||||||
|
\begin{tabular}{l r r r}
|
||||||
|
\hline
|
||||||
|
state & init halfwidth & final halfwidth & ratio \\
|
||||||
|
\hline
|
||||||
|
$n$ & $0.010$ & $0.083$ & 8.3$\times$ expand \\
|
||||||
|
$C_1 \dots C_6$ & varies & varies & 160--200$\times$ expand \\
|
||||||
|
$T_f$ & 0.1~\unit{\kelvin} & 2.08~\unit{\kelvin} & 21$\times$ expand \\
|
||||||
|
$\boldsymbol{T_c}$ & $\mathbf{0.1}$~\unit{\kelvin} & $\mathbf{0.033}$~\unit{\kelvin} & $\mathbf{0.33\times}$ \textbf{contract} \\
|
||||||
|
$T_{\mathrm{cold}}$ & 0.1~\unit{\kelvin} & 1.47~\unit{\kelvin} & 15$\times$ expand \\
|
||||||
|
\hline
|
||||||
|
\end{tabular}
|
||||||
|
\end{center}
|
||||||
|
|
||||||
|
$T_c$ \emph{contracts} — LQR drags the regulated direction toward
|
||||||
|
setpoint faster than disturbance can push it. Uncontrolled states
|
||||||
|
drift. Precursor expansion ($\sim$$200\times$) is immaterial for
|
||||||
|
safety (no predicate uses them).
|
||||||
|
|
||||||
|
\subsection*{Part 8: Lyapunov barrier attempt}
|
||||||
|
|
||||||
|
A reach tube is a numerical certificate: compute, check. A
|
||||||
|
\emph{barrier certificate} is analytic: produce a function $B(x)$ such
|
||||||
|
that $B \leq 0$ on the initial set, $B > 0$ on the unsafe set, and
|
||||||
|
$\dot B \leq 0$ when $B = 0$. Forward-invariance of the safe set
|
||||||
|
follows from the dynamics without iterating them.
|
||||||
|
|
||||||
|
For linear systems with bounded disturbance, the canonical candidate
|
||||||
|
is a Lyapunov quadratic.
|
||||||
|
|
||||||
|
\begin{derivation}
|
||||||
|
Let $V(\delta x) = \delta x^\top P \delta x$ with $P \succ 0$ solving
|
||||||
|
$A_{\mathrm{cl}}^\top P + P A_{\mathrm{cl}} + \bar Q = 0$. Along
|
||||||
|
trajectories of $\dot{\delta x} = A_{\mathrm{cl}} \delta x + B_w w$:
|
||||||
|
\begin{align}
|
||||||
|
\dot V &= -\delta x^\top \bar Q \delta x + 2 \delta x^\top P B_w w.
|
||||||
|
\end{align}
|
||||||
|
Bound each piece. The dissipation term using the generalized
|
||||||
|
eigenvalue
|
||||||
|
$\mu = \lambda_{\min}\bigl(P^{-1/2} \bar Q P^{-1/2}\bigr)$:
|
||||||
|
$$-\delta x^\top \bar Q \delta x \leq -\mu V.$$
|
||||||
|
The disturbance kick, Cauchy-Schwarz in the $P$-metric:
|
||||||
|
$$2 \delta x^\top P B_w w \leq 2 \bar w \sqrt{B_w^\top P B_w}\sqrt{V}.$$
|
||||||
|
Combining:
|
||||||
|
$$\dot V \leq -\mu V + 2 \bar w \sqrt{B_w^\top P B_w}\sqrt{V}.$$
|
||||||
|
Set $s = \sqrt{V}$, requiring $\dot s \leq 0$:
|
||||||
|
$$s \geq \frac{2 \bar w \sqrt{B_w^\top P B_w}}{\mu}
|
||||||
|
\quad \Longleftrightarrow \quad V \geq \left(\frac{2 \bar w \sqrt{B_w^\top P B_w}}{\mu}\right)^2 =: c_{\mathrm{inv}}.$$
|
||||||
|
The sub-level set $\{ V \leq c_{\mathrm{inv}} \}$ is forward-invariant
|
||||||
|
under any admissible disturbance.
|
||||||
|
\end{derivation}
|
||||||
|
|
||||||
|
So the recipe is: solve Lyapunov, compute $c_{\mathrm{inv}}$, compute
|
||||||
|
the worst-case deviation of $T_c$ (or any linear functional of state)
|
||||||
|
on the resulting ellipsoid, and check whether it fits inside the
|
||||||
|
safety slab.
|
||||||
|
|
||||||
|
\begin{derivation}
|
||||||
|
Max of a linear functional $a^\top \delta x$ over the ellipsoid
|
||||||
|
$\{\delta x : \delta x^\top P \delta x \leq \gamma\}$:
|
||||||
|
$$\max a^\top \delta x = \sqrt{\gamma \cdot a^\top P^{-1} a}.$$
|
||||||
|
(Lagrange multipliers: the maximum is achieved at
|
||||||
|
$\delta x = \sqrt{\gamma / a^\top P^{-1} a} \cdot P^{-1} a$.)
|
||||||
|
\end{derivation}
|
||||||
|
|
||||||
|
\textbf{Result:} the best quadratic barrier — across a sweep of
|
||||||
|
$\bar Q(9,9)$ from 10 to $10^6$ — allows max $|T_c - T_{c0}| \approx 33$~\unit{\kelvin},
|
||||||
|
more than six times the 5~\unit{\kelvin} safety band. On the hard
|
||||||
|
safety halfspaces (\texttt{inv2\_holds}), it says $n$ could deviate by
|
||||||
|
$\pm 1242$~$\times$ nominal — physically meaningless.
|
||||||
|
|
||||||
|
\begin{limitation}
|
||||||
|
\textbf{The quadratic Lyapunov barrier fails on this plant. This is
|
||||||
|
a structural finding, not a tuning failure.}
|
||||||
|
|
||||||
|
The safety spec is anisotropic: a thin slab $|T_c| \leq 5$~\unit{\kelvin}
|
||||||
|
in a 10-D state space where $T_c$ is one direction and the other nine
|
||||||
|
are precursors / fuel temperature / cold-leg. Any
|
||||||
|
quadratic $V$ yields axis-uniform ellipsoidal level sets. To contain
|
||||||
|
the disturbance-driven invariant set, the ellipsoid must be \emph{fat
|
||||||
|
in the slow directions} (precursors, where Lyapunov dissipation is
|
||||||
|
weakest). Projecting that fat ellipsoid back onto $T_c$ yields a
|
||||||
|
range much larger than the physical $T_c$ behavior.
|
||||||
|
|
||||||
|
Confirmed the issue is not controller tuning by running the barrier
|
||||||
|
open-loop (no LQR): the bound becomes $\pm 27{,}000{,}000\,n$. LQR
|
||||||
|
helps by $\sim$20{,}000$\times$, but still leaves a 3-4 order-of-magnitude
|
||||||
|
gap to usability. The ceiling is plant anisotropy: prompt-neutron
|
||||||
|
timescale $\Lambda = 10^{-4}$~\unit{\second} vs.\ thermal timescales
|
||||||
|
$\sim$10--100~\unit{\second}, a factor-of-$10^4$ spread that forces
|
||||||
|
$P$ to be ill-conditioned regardless of controller design.
|
||||||
|
|
||||||
|
\textbf{Fix:} non-quadratic barrier. Two candidate approaches:
|
||||||
|
\begin{itemize}
|
||||||
|
\item \textbf{Polytopic barrier:} $B(\delta x) = \max_i (a_i^\top \delta x - b_i)$.
|
||||||
|
Piecewise-linear, can match a slab shape directly.
|
||||||
|
Requires polyhedral-set tooling.
|
||||||
|
\item \textbf{SOS polynomial barrier:} $B$ a polynomial of
|
||||||
|
degree $\geq 4$, coefficients found by sum-of-squares
|
||||||
|
optimization. Requires an SOS solver (Mosek, SDPT3).
|
||||||
|
\end{itemize}
|
||||||
|
Neither is implemented; both are in-scope for the thesis chapter.
|
||||||
|
The quadratic-Lyapunov failure, stated quantitatively, is the clean
|
||||||
|
motivation.
|
||||||
|
\end{limitation}
|
||||||
|
|
||||||
|
\subsection*{Session result}
|
||||||
|
|
||||||
|
Artifacts landing, in order:
|
||||||
|
\begin{enumerate}
|
||||||
|
\item \texttt{plant-model/controllers/ctrl\_shutdown.m},
|
||||||
|
\texttt{ctrl\_scram.m}, \texttt{ctrl\_heatup.m} (three new
|
||||||
|
controllers).
|
||||||
|
\item \texttt{plant-model/pke\_linearize.m} and
|
||||||
|
\texttt{plant-model/test\_linearize.m}.
|
||||||
|
\item \texttt{plant-model/controllers/ctrl\_operation\_lqr.m}.
|
||||||
|
\item \texttt{plant-model/main\_mode\_sweep.m}.
|
||||||
|
\item \texttt{reachability/reach\_linear.m} and
|
||||||
|
\texttt{reachability/reach\_operation.m}.
|
||||||
|
\item \texttt{reachability/barrier\_lyapunov.m}.
|
||||||
|
\item Eleven figures in \texttt{docs/figures/}.
|
||||||
|
\end{enumerate}
|
||||||
|
|
||||||
|
Key claims established:
|
||||||
|
\begin{itemize}
|
||||||
|
\item LQR closed-loop operation mode satisfies the safety obligation
|
||||||
|
under $\pm 15\%$ $Q_{\mathrm{sg}}$ variation (approximate,
|
||||||
|
via linear reach).
|
||||||
|
\item Quadratic Lyapunov barrier insufficient on this plant
|
||||||
|
(open-loop or closed-loop). Need polytopic or SOS.
|
||||||
|
\item LQR adds zero state, dominates plain P by $\sim 5\times$
|
||||||
|
on tracking.
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
\subsection*{Limitations recorded at close}
|
||||||
|
\label{sec:limitations-17}
|
||||||
|
|
||||||
|
\begin{limitation}
|
||||||
|
All reach results are \emph{approximate}, not sound: they are reach
|
||||||
|
tubes of the \emph{linearized} closed-loop. The linearization error
|
||||||
|
— the gap between $f_{\mathrm{nl}}(x, u)$ and $(A x + B u)$ — is not
|
||||||
|
propagated into the tube. For a real safety claim, either
|
||||||
|
(a)~nonlinear reach directly or (b)~linear reach plus Taylor-remainder
|
||||||
|
inflation. Neither is done.
|
||||||
|
\end{limitation}
|
||||||
|
|
||||||
|
\begin{limitation}
|
||||||
|
\label{sub:alpha-drift}
|
||||||
|
The feedback-linearization in \texttt{ctrl\_heatup} assumes exact
|
||||||
|
$\alpha_f, \alpha_c$. Real PWRs have $\alpha$ drift: $\sim$20\%
|
||||||
|
over burnup; $\sim$10$\times$ across soluble-boron dilution; xenon
|
||||||
|
transients worth 2--3~\$. Robust version would treat $\alpha$ as
|
||||||
|
parametric interval and propagate through reach.
|
||||||
|
\end{limitation}
|
||||||
|
|
||||||
|
\begin{limitation}
|
||||||
|
Saturation in \texttt{ctrl\_heatup} is formally a 3-mode piecewise-affine
|
||||||
|
sub-system. Operation-mode LQR is dormant-saturated (trivially); a
|
||||||
|
rigorous heatup reach would need explicit region decomposition or a
|
||||||
|
dormancy proof.
|
||||||
|
\end{limitation}
|
||||||
|
|
||||||
|
\begin{limitation}
|
||||||
|
The model's $\alpha_f, \alpha_c$ are linearized at the hot full-power
|
||||||
|
operating point. Trust region is $\pm 50$~\unit{\celsius} around $x^\star$.
|
||||||
|
True cold shutdown ($\sim$50~\unit{\celsius}) is out of scope. What
|
||||||
|
the DRC calls \texttt{q\_shutdown} we interpret as hot standby
|
||||||
|
($\sim$290~\unit{\celsius}).
|
||||||
|
\end{limitation}
|
||||||
|
|
||||||
|
\subsection*{Open at close}
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item Nonlinear reach, at minimum on one mode, to retire the
|
||||||
|
soundness asterisk.
|
||||||
|
\item Polytopic or SOS barrier to retire the analytic-certificate
|
||||||
|
asterisk.
|
||||||
|
\item Parametric $\alpha$ uncertainty in the reach machinery.
|
||||||
|
\item Heatup reach — ramped reference needs LTV or nonlinear
|
||||||
|
treatment.
|
||||||
|
\item Shutdown + scram reach (trivial forever-invariance /
|
||||||
|
bounded-time respectively, but not yet done).
|
||||||
|
\end{itemize}
|
||||||
@ -0,0 +1,472 @@
|
|||||||
|
% ---------------------------------------------------------------------------
|
||||||
|
% 2026-04-20 — Predicates restructure, mode taxonomy, Julia nonlinear reach
|
||||||
|
% Deep / A-style invention-log entry.
|
||||||
|
% ---------------------------------------------------------------------------
|
||||||
|
|
||||||
|
\session{2026-04-20}{afternoon session, $\sim$3 hr}{Walk through the
|
||||||
|
2026-04-17 work with a critical eye, surface three structural errors
|
||||||
|
hiding in it, restructure the predicates JSON to fix them, formalize
|
||||||
|
the per-mode reach-obligation taxonomy (equilibrium vs.\ transition),
|
||||||
|
and take the first swing at Julia nonlinear reach. The Julia reach
|
||||||
|
framework works but is limited to $\sim$10-second horizons by
|
||||||
|
prompt-neutron stiffness. The remedy (singular-perturbation reduction)
|
||||||
|
is identified and deferred.}
|
||||||
|
|
||||||
|
\section{2026-04-20 — Predicates restructure, mode taxonomy, nonlinear reach}
|
||||||
|
\label{sec:20260420-afternoon}
|
||||||
|
|
||||||
|
\subsection*{How this session started}
|
||||||
|
|
||||||
|
Walking through the 2026-04-17 work slide-by-slide with a critical
|
||||||
|
reviewer looking for inconsistencies. Each section of the linear
|
||||||
|
reach story had a latent bug the first pass missed; by the end of the
|
||||||
|
walkthrough three structural errors were exposed and fixed.
|
||||||
|
|
||||||
|
\subsection*{Error 1: the barrier was checking the wrong invariant}
|
||||||
|
|
||||||
|
The 2026-04-17 barrier code compared the Lyapunov-ellipsoid reach
|
||||||
|
against a \emph{symmetric} slab $|T_c - T_{c0}| \leq 2.78$~\unit{\celsius}
|
||||||
|
— the operational deadband \texttt{t\_avg\_in\_range}. But that
|
||||||
|
predicate is used by the DRC for \emph{mode transitions}: crossing it
|
||||||
|
triggers heatup$\to$operation, not damage. The barrier should be
|
||||||
|
checking the \emph{safety limits} — one-sided halfspaces reflecting
|
||||||
|
actual trip setpoints and physical damage thresholds. These are not
|
||||||
|
symmetric:
|
||||||
|
\begin{itemize}
|
||||||
|
\item $T_f \leq 1200$~\unit{\celsius} (fuel centerline design limit).
|
||||||
|
\item $T_c \leq 320$~\unit{\celsius} (high-$T_{\mathrm{avg}}$ trip).
|
||||||
|
\item $T_c \geq 280$~\unit{\celsius} (low-$T_{\mathrm{avg}}$ trip).
|
||||||
|
\item $n \leq 1.15$ (high-flux trip).
|
||||||
|
\item $n \geq 0.15$ (operation range only).
|
||||||
|
\item $T_{\mathrm{cold}} \leq T_{\mathrm{cold0}} + 15$ (subcooling).
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
\begin{decision}
|
||||||
|
Restructure \texttt{reachability/predicates.json} into three categories:
|
||||||
|
\begin{description}
|
||||||
|
\item[\texttt{operational\_deadbands}] soft bands used by DRC for
|
||||||
|
mode switching. \texttt{t\_avg\_above\_min},
|
||||||
|
\texttt{t\_avg\_in\_range}, \texttt{p\_above\_crit}.
|
||||||
|
\item[\texttt{safety\_limits}] hard one-sided halfspaces. Six rows
|
||||||
|
as listed above, plus new heatup-rate ones (see Error 3).
|
||||||
|
\item[\texttt{mode\_invariants}] \texttt{inv1\_holds},
|
||||||
|
\texttt{inv2\_holds}: declared as \emph{conjunctions} of
|
||||||
|
named safety limits, so changing a limit propagates.
|
||||||
|
\end{description}
|
||||||
|
Safety proofs target \texttt{mode\_invariants}; DRC transitions
|
||||||
|
target \texttt{operational\_deadbands}. The distinction is
|
||||||
|
load-bearing.
|
||||||
|
\end{decision}
|
||||||
|
|
||||||
|
\subsection*{Error 2: the "2364$\times$ bound" should have included LQR, and didn't tell us}
|
||||||
|
|
||||||
|
After the restructure, re-running \texttt{barrier\_lyapunov.m} against
|
||||||
|
\texttt{inv2\_holds} produced:
|
||||||
|
|
||||||
|
\begin{lstlisting}[style=terminal]
|
||||||
|
=== Lyapunov barrier vs inv2_holds halfspaces ===
|
||||||
|
[fuel_centerline] max a'dx = 2244.1 margin = -1372.4 too loose
|
||||||
|
[t_avg_high_trip] max a'dx = 33.3 margin = -21.6 too loose
|
||||||
|
[n_high_trip] max a'dx = 1242.0 margin = -1242.0 too loose (!!)
|
||||||
|
[cold_leg_subcooled] max a'dx = 150.7 margin = -135.7 too loose
|
||||||
|
\end{lstlisting}
|
||||||
|
|
||||||
|
The $n_{\mathrm{high\_trip}}$ row was the stopper: the barrier says
|
||||||
|
$n$ could deviate by $\pm 1242\times$ nominal. Physically meaningless.
|
||||||
|
Fair enough to ask: is this because \emph{LQR isn't in the
|
||||||
|
barrier computation?} The code uses $A_{\mathrm{cl}} = A - BK$, but
|
||||||
|
that detail was buried and Dane rightly flagged it.
|
||||||
|
|
||||||
|
\begin{decision}
|
||||||
|
\textbf{Write \texttt{barrier\_compare\_OL\_CL.m}} to explicitly
|
||||||
|
measure LQR's contribution to the barrier. Same Qbar, same
|
||||||
|
disturbance envelope, two runs: open-loop $A$ vs.\ closed-loop
|
||||||
|
$A_{\mathrm{cl}} = A - BK$.
|
||||||
|
\end{decision}
|
||||||
|
|
||||||
|
Result (\cref{fig:ol-vs-cl}):
|
||||||
|
|
||||||
|
\begin{figure}[h]
|
||||||
|
\centering
|
||||||
|
\begin{tabular}{lrrr}
|
||||||
|
\hline
|
||||||
|
halfspace & open-loop & LQR closed-loop & ratio CL/OL \\
|
||||||
|
\hline
|
||||||
|
\texttt{fuel\_centerline} & 26{,}941{,}862 K & 1{,}137 K & $4.2 \times 10^{-5}$ \\
|
||||||
|
\texttt{t\_avg\_high\_trip} & 788{,}220 K & 33.2 K & $4.2 \times 10^{-5}$ \\
|
||||||
|
\texttt{n\_high\_trip} & 27{,}379{,}931 & 1{,}242 & $4.5 \times 10^{-5}$ \\
|
||||||
|
\texttt{cold\_leg\_subcooled} & 1{,}806{,}166 K& 77.8 K & $4.3 \times 10^{-5}$ \\
|
||||||
|
$\gamma$ (ellipsoid level) & $1.04 \times 10^{13}$ & $1.85 \times 10^{4}$ & $1.78 \times 10^{-9}$ \\
|
||||||
|
\hline
|
||||||
|
\end{tabular}
|
||||||
|
\caption{Open-loop vs.\ LQR closed-loop Lyapunov barrier bounds on
|
||||||
|
each \texttt{inv2\_holds} halfspace, same $\bar Q$ and same
|
||||||
|
disturbance envelope. LQR improves every bound by a uniform
|
||||||
|
$\sim$20{,}000$\times$, but the bounds remain physically absurd.
|
||||||
|
The open-loop version says $n$ could deviate by 27~million; the
|
||||||
|
closed-loop says 1{,}242. Both unusable. The ceiling is plant
|
||||||
|
anisotropy ($\Lambda/\tau_{\mathrm{thermal}} \sim 10^{-4}$ timescale
|
||||||
|
ratio), not controller tuning. Generated by
|
||||||
|
\texttt{barrier\_compare\_OL\_CL.m}.}
|
||||||
|
\label{fig:ol-vs-cl}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
\begin{derivation}
|
||||||
|
Why the eigenvalues barely move but $\gamma$ changes 9 orders of
|
||||||
|
magnitude:
|
||||||
|
\begin{itemize}
|
||||||
|
\item $\max \Re(\mathrm{eig}\,A) = -0.0125$ — slowest thermal mode.
|
||||||
|
\item $\max \Re(\mathrm{eig}\,A_{\mathrm{cl}}) = -0.0124$ — virtually
|
||||||
|
identical. LQR cannot speed up the slowest mode past what
|
||||||
|
physics allows.
|
||||||
|
\item But $\gamma = c_{\mathrm{inv}} \propto (w_{\mathrm{bar}} \sqrt{B_w^\top P B_w} / \mu)^2$,
|
||||||
|
and $P$ changes \emph{shape} between OL and CL: LQR makes the
|
||||||
|
disturbance penetrate $V$ far less. $B_w^\top P B_w$ drops
|
||||||
|
substantially under LQR, $\mu$ stays similar. Ratio squared
|
||||||
|
$\to$ nine orders of magnitude.
|
||||||
|
\end{itemize}
|
||||||
|
\end{derivation}
|
||||||
|
|
||||||
|
\begin{limitation}
|
||||||
|
Quadratic Lyapunov barrier is insufficient for this plant regardless
|
||||||
|
of LQR tuning. Root cause: the plant's natural timescale spread
|
||||||
|
(prompt-neutron $\Lambda = 10^{-4}$~\unit{\second} vs.\ thermal $\sim 10$~\unit{\second})
|
||||||
|
forces any Lyapunov $P$ to be ill-conditioned by $\sim 10^4$. The
|
||||||
|
resulting ellipsoid stretches absurdly in the fast directions when
|
||||||
|
projected to physical coordinates. Fix:\ non-quadratic barrier
|
||||||
|
(polytopic or SOS). Neither implemented yet.
|
||||||
|
\end{limitation}
|
||||||
|
|
||||||
|
\subsection*{Error 3: the heatup-rate invariant IS expressible as a halfspace}
|
||||||
|
|
||||||
|
In the first version of \texttt{predicates.json}, I put a comment on
|
||||||
|
\texttt{inv1\_holds} claiming a ramp-rate constraint could not be
|
||||||
|
written as a state halfspace without state augmentation. That was
|
||||||
|
wrong.
|
||||||
|
|
||||||
|
\begin{derivation}
|
||||||
|
From \texttt{pke\_th\_rhs.m}, $\dot T_c$ is linear in
|
||||||
|
$(T_f, T_c, T_{\mathrm{cold}})$:
|
||||||
|
$$\dot T_c = \frac{hA(T_f - T_c) - 2 W c_c (T_c - T_{\mathrm{cold}})}{M_c c_c}
|
||||||
|
= a_f T_f + a_c T_c + a_{\mathrm{cold}} T_{\mathrm{cold}}$$
|
||||||
|
with
|
||||||
|
\begin{align*}
|
||||||
|
a_f &= \frac{hA}{M_c c_c} = \frac{5 \times 10^7}{1.09 \times 10^8} = +0.4587~\unit{\per\second} \\
|
||||||
|
a_c &= -\frac{hA + 2 W c_c}{M_c c_c} = -\frac{1.045 \times 10^8}{1.09 \times 10^8} = -0.9587~\unit{\per\second} \\
|
||||||
|
a_{\mathrm{cold}} &= \frac{2 W c_c}{M_c c_c} = \frac{5.45 \times 10^7}{1.09 \times 10^8} = +0.5000~\unit{\per\second}
|
||||||
|
\end{align*}
|
||||||
|
Coefficients sum to zero exactly: at thermal equilibrium
|
||||||
|
($T_f = T_c = T_{\mathrm{cold}}$), $\dot T_c = 0$.
|
||||||
|
|
||||||
|
A ramp-rate constraint $|\dot T_c| \leq r_{\max}$ is therefore two
|
||||||
|
halfspaces over the 10-state vector, with three nonzero coefficients
|
||||||
|
each (on indices $8, 9, 10$):
|
||||||
|
\begin{align*}
|
||||||
|
+a_f T_f + a_c T_c + a_{\mathrm{cold}} T_{\mathrm{cold}} &\leq r_{\max} \\
|
||||||
|
-a_f T_f - a_c T_c - a_{\mathrm{cold}} T_{\mathrm{cold}} &\leq r_{\max}
|
||||||
|
\end{align*}
|
||||||
|
Clean polyhedron; no augmentation. The confusion came from conflating
|
||||||
|
this with rate constraints on \emph{non-linearly-driven} states — e.g.\
|
||||||
|
$|\dot n|$ involves $(\rho - \beta) n / \Lambda$, which is nonlinear,
|
||||||
|
and \emph{that} would need augmentation. For the coolant temperature
|
||||||
|
rate, the thermal-hydraulic subsystem is linear-in-state and the
|
||||||
|
halfspace drops out directly.
|
||||||
|
\end{derivation}
|
||||||
|
|
||||||
|
$r_{\max}$ chosen as 50~\unit{\celsius\per\hour} (tech-spec nominal is
|
||||||
|
28~\unit{\celsius\per\hour}; add budget for transient overshoot).
|
||||||
|
\texttt{plant-model/test\_heatup\_rate.m} measures the actual peak
|
||||||
|
rate from the current \texttt{ctrl\_heatup.m} controller:
|
||||||
|
|
||||||
|
\begin{lstlisting}[style=terminal]
|
||||||
|
=== Heatup rate statistics ===
|
||||||
|
max dT_c/dt: 0.0135 C/s = 48.5 C/hr
|
||||||
|
min dT_c/dt: 0.0000 C/s = 0.0 C/hr
|
||||||
|
rate limit: +/- 0.01389 C/s = +/- 50.0 C/hr
|
||||||
|
violates upper? false
|
||||||
|
violates lower? false
|
||||||
|
\end{lstlisting}
|
||||||
|
|
||||||
|
Right at 50~\unit{\celsius\per\hour} peak during mid-ramp — barely
|
||||||
|
inside the budget, and \emph{70\% above tech-spec nominal}. Would
|
||||||
|
fail a strict 28~\unit{\celsius\per\hour} interpretation. Documented
|
||||||
|
as an open controller-tuning item.
|
||||||
|
|
||||||
|
\subsection*{The mode-obligation taxonomy}
|
||||||
|
|
||||||
|
The key conceptual move of the session. Walking through the reach
|
||||||
|
code, Dane observed that the per-mode reach obligations aren't all the
|
||||||
|
same shape. They split cleanly in two:
|
||||||
|
|
||||||
|
\begin{decision}
|
||||||
|
\textbf{Two kinds of mode, two kinds of obligation.}
|
||||||
|
|
||||||
|
\textbf{Equilibrium modes} (\texttt{q\_operation}, \texttt{q\_shutdown}):
|
||||||
|
plant is parked at a setpoint. Obligation is forever-invariance under
|
||||||
|
bounded disturbance. External disturbance matters — it's the thing
|
||||||
|
that could push the state out. \emph{This is what the 2026-04-17
|
||||||
|
operation-mode reach actually proves} (up to the linearization
|
||||||
|
caveat).
|
||||||
|
|
||||||
|
\textbf{Transition modes} (\texttt{q\_heatup}, \texttt{q\_scram}):
|
||||||
|
plant is being driven from one mode's territory to another's.
|
||||||
|
Obligation is \emph{reach-avoid}: from $X_{\mathrm{entry}}$, reach
|
||||||
|
$X_{\mathrm{exit}}$ within bounded time $[T_{\min}, T_{\max}]$, while
|
||||||
|
maintaining $\mathrm{inv}_{\mathrm{mode}}$ along the way. External
|
||||||
|
disturbance in these modes is essentially zero
|
||||||
|
($Q_{\mathrm{sg}} = 0$ during heatup, rods slam on scram). The only
|
||||||
|
``disturbance'' is parametric uncertainty in the plant coefficients,
|
||||||
|
which we defer.
|
||||||
|
\end{decision}
|
||||||
|
|
||||||
|
The point of per-mode reach is \emph{not} generic disturbance
|
||||||
|
rejection — it's to prove that the DRC's discrete transitions
|
||||||
|
physically fire in finite time on the real plant. The reach tube is
|
||||||
|
the artifact that transfers discrete correctness
|
||||||
|
(\texttt{ltlsynt}-guaranteed) to physical correctness.
|
||||||
|
|
||||||
|
Formal claim per transition mode (heatup example):
|
||||||
|
\begin{align}
|
||||||
|
&\text{Given } x(0) \in X_{\mathrm{entry}}(\text{heatup}), \notag \\
|
||||||
|
&\text{applying } u = \texttt{ctrl\_heatup}(t, x, \mathrm{plant}, \mathrm{ref}), \notag \\
|
||||||
|
&\text{with } Q_{\mathrm{sg}} = 0, \notag \\
|
||||||
|
&\text{we show } \exists\, T \in [T_{\min}, T_{\max}] \text{ s.t.} \notag \\
|
||||||
|
&\quad x(T) \in X_{\mathrm{exit}}(\text{heatup}) \quad \text{[reach]} \notag \\
|
||||||
|
&\quad x(t) \in \mathrm{inv1\_holds} \quad \forall t \in [0, T] \quad \text{[avoid]}
|
||||||
|
\end{align}
|
||||||
|
|
||||||
|
The entry/exit sets and time budgets need machine-readable
|
||||||
|
definitions. Added a \texttt{mode\_boundaries} block to
|
||||||
|
\texttt{predicates.json}:
|
||||||
|
\begin{itemize}
|
||||||
|
\item Each mode's \texttt{X\_entry\_polytope} as box bounds per state.
|
||||||
|
\item Each mode's \texttt{X\_exit\_predicate} as a reference to named
|
||||||
|
predicates.
|
||||||
|
\item Each mode's \texttt{T\_min}, \texttt{T\_max} (\texttt{null} for
|
||||||
|
equilibrium modes).
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
Values are engineering-reasonable placeholders:
|
||||||
|
\texttt{T\_max(heatup)} = 5~\unit{\hour} (tech-spec 28~\unit{\celsius/\hour}
|
||||||
|
over 60~\unit{\degreeFahrenheit} span is 1.19~\unit{\hour} nominal, 4$\times$
|
||||||
|
margin). \texttt{T\_min(heatup)} = 2~\unit{\hour} 9~\unit{\minute} (any
|
||||||
|
faster violates the rate invariant). \texttt{T\_max(scram)} = 60~\unit{\second}
|
||||||
|
(NRC expects seconds; 60 is generous for idealized rod insertion).
|
||||||
|
|
||||||
|
\begin{limitation}
|
||||||
|
All mode-boundary values are engineering guesses. Thesis defense
|
||||||
|
will require real tech-spec numbers pinned to a specific plant.
|
||||||
|
Flagged in the JSON as \texttt{\_placeholder\_warning}.
|
||||||
|
\end{limitation}
|
||||||
|
|
||||||
|
\subsection*{WALKTHROUGH.md — standalone reach documentation}
|
||||||
|
|
||||||
|
With the mode-obligation taxonomy, predicate restructure, and barrier
|
||||||
|
findings in hand, wrote \texttt{reachability/WALKTHROUGH.md} as a
|
||||||
|
standalone document. Captures the taxonomy table verbatim, per-mode
|
||||||
|
obligation definitions, code walkthrough, per-halfspace results, and
|
||||||
|
every known limitation.
|
||||||
|
|
||||||
|
\apass{WALKTHROUGH.md should probably be folded into a journal-style
|
||||||
|
chapter at some point, but for now it's the external-facing doc people
|
||||||
|
read first. Keep it in sync when structural things change.}
|
||||||
|
|
||||||
|
\subsection*{Julia nonlinear reach — first attempt, partial success}
|
||||||
|
|
||||||
|
With linear work consolidated, turned to the real soundness question.
|
||||||
|
The linear reach proves the LQR closed-loop is safe, \emph{if} we
|
||||||
|
trust the linearization. To close the gap, we need nonlinear reach.
|
||||||
|
|
||||||
|
Framework choice: \texttt{ReachabilityAnalysis.jl} with TMJets (Taylor
|
||||||
|
model integration). Already have Julia scaffolding in
|
||||||
|
\texttt{julia-port/} and know the package.
|
||||||
|
|
||||||
|
Target: heatup mode reach, saturation disabled (simplifying
|
||||||
|
assumption to establish the framework; will add back saturation as
|
||||||
|
a hybrid sub-mode later).
|
||||||
|
|
||||||
|
\subsubsection*{The first three attempts failed}
|
||||||
|
|
||||||
|
\begin{deadend}
|
||||||
|
\textbf{Attempt 1:} naive RHS with \texttt{plant} as a function
|
||||||
|
argument. Fails immediately with
|
||||||
|
\texttt{MethodError: setindex!(::Taylor1\{Float64\}, ::TaylorModel1\{...\}, ...)} —
|
||||||
|
Taylor model arithmetic needs the RHS in a specific form. Need
|
||||||
|
\texttt{@taylorize}.
|
||||||
|
|
||||||
|
\textbf{Attempt 2:} add \texttt{@taylorize} to the function. Same error.
|
||||||
|
Missing piece: the macro only handles \emph{constants} in arithmetic,
|
||||||
|
not struct-field accesses. Need all plant parameters as \texttt{const}
|
||||||
|
globals.
|
||||||
|
|
||||||
|
\textbf{Attempt 3:} inline all constants. Still fails. Running out
|
||||||
|
of ideas; then noticed that \texttt{min()} inside the body (for the
|
||||||
|
ramp-reference clamp) is non-smooth — Taylor models can't handle
|
||||||
|
non-smooth operations. Also: the raw time argument \texttt{t} in the
|
||||||
|
signature was interacting badly with TMJets' internal time parameter.
|
||||||
|
\end{deadend}
|
||||||
|
|
||||||
|
\begin{decision}
|
||||||
|
\textbf{Attempt 4 (landed):} three changes together.
|
||||||
|
\begin{enumerate}
|
||||||
|
\item All plant parameters as \texttt{const} at module scope.
|
||||||
|
\item \texttt{@taylorize} on the RHS, all arithmetic scalar (loops
|
||||||
|
unrolled, no function calls in the hot path).
|
||||||
|
\item Time carried as an \emph{augmented state} $x_{11}$ with
|
||||||
|
$\dot x_{11} = 1$. Instead of $T_{\mathrm{ref}}(t)$, the RHS
|
||||||
|
references $T_{\mathrm{ref}}(x_{11})$ with no \texttt{min()}
|
||||||
|
— valid while the ramp hasn't hit the target clamp, which is
|
||||||
|
true for our probe horizons.
|
||||||
|
\end{enumerate}
|
||||||
|
\end{decision}
|
||||||
|
|
||||||
|
The final RHS body (see
|
||||||
|
\texttt{julia-port/scripts/reach\_heatup\_nonlinear.jl}):
|
||||||
|
|
||||||
|
\begin{lstlisting}[language=Julia,caption={Taylor-model-aware heatup RHS}]
|
||||||
|
@taylorize function rhs_heatup_taylor!(dx, x, p, t)
|
||||||
|
rho = KP_HEATUP * (T_STANDBY + RAMP_RATE_CS * x[11] - x[9])
|
||||||
|
|
||||||
|
dx[1] = (rho - BETA) / LAMBDA * x[1] +
|
||||||
|
LAM_1*x[2] + LAM_2*x[3] + LAM_3*x[4] +
|
||||||
|
LAM_4*x[5] + LAM_5*x[6] + LAM_6*x[7]
|
||||||
|
dx[2] = (BETA_1 / LAMBDA) * x[1] - LAM_1 * x[2]
|
||||||
|
dx[3] = (BETA_2 / LAMBDA) * x[1] - LAM_2 * x[3]
|
||||||
|
dx[4] = (BETA_3 / LAMBDA) * x[1] - LAM_3 * x[4]
|
||||||
|
dx[5] = (BETA_4 / LAMBDA) * x[1] - LAM_4 * x[5]
|
||||||
|
dx[6] = (BETA_5 / LAMBDA) * x[1] - LAM_5 * x[6]
|
||||||
|
dx[7] = (BETA_6 / LAMBDA) * x[1] - LAM_6 * x[7]
|
||||||
|
|
||||||
|
dx[8] = (P0 * x[1] - HA * (x[8] - x[9])) / (M_F * C_F)
|
||||||
|
dx[9] = (HA * (x[8] - x[9]) - 2 * W_M * C_C * (x[9] - x[10])) / (M_C * C_C)
|
||||||
|
dx[10] = (2 * W_M * C_C * (x[9] - x[10])) / (M_SG * C_C)
|
||||||
|
|
||||||
|
dx[11] = one(x[1]) # d(time)/dt = 1, in the right Taylor type
|
||||||
|
return nothing
|
||||||
|
end
|
||||||
|
\end{lstlisting}
|
||||||
|
|
||||||
|
Probes at $T = 10, 60, 300$~\unit{\second}:
|
||||||
|
|
||||||
|
\begin{lstlisting}[style=terminal]
|
||||||
|
--- Probe T = 10.0 s ---
|
||||||
|
TMJets: 10583 reach-sets
|
||||||
|
t reached: 10.0 s of 10.0
|
||||||
|
n envelope: [-0.0005178, 0.005015]
|
||||||
|
T_f envelope: [274.46, 295.01] C
|
||||||
|
T_c envelope: [274.45, 295.00] C
|
||||||
|
T_cold env: [270.00, 287.16] C
|
||||||
|
|
||||||
|
--- Probe T = 60.0 s ---
|
||||||
|
Warning: Maximum number of integration steps reached; exiting.
|
||||||
|
TMJets: 50000 reach-sets
|
||||||
|
FAILED: AssertionError: radius must be nonnegative but is [NaN, NaN, ...]
|
||||||
|
|
||||||
|
--- Probe T = 300.0 s ---
|
||||||
|
(same as 60s)
|
||||||
|
\end{lstlisting}
|
||||||
|
|
||||||
|
\textbf{10 seconds works}; 60 seconds onward exhaust the step budget
|
||||||
|
and then propagate NaN. The 10-second envelope is sound — the
|
||||||
|
$n$-envelope going slightly negative is over-approximation tax
|
||||||
|
(physically impossible, numerically correct for a box hull).
|
||||||
|
|
||||||
|
\begin{derivation}
|
||||||
|
Step-size analysis. TMJets' adaptive stepper shrinks $\Delta t$ to
|
||||||
|
resolve the fastest active mode. Prompt-neutron timescale is
|
||||||
|
$\Lambda = 10^{-4}$~\unit{\second}. For the Taylor remainder
|
||||||
|
(\texttt{abstol=1e-10}) to be bounded, the stepper needs
|
||||||
|
$\Delta t \lesssim 10^{-3}$~\unit{\second}. Over a 10~\unit{\second}
|
||||||
|
horizon that's $\sim 10{,}000$ steps — consistent with the observed
|
||||||
|
10{,}583.
|
||||||
|
|
||||||
|
Extrapolate: a 5-hour heatup reach would need $\sim 1.8 \times 10^7$
|
||||||
|
steps at this cadence. Totally infeasible, and we'd hit numerical
|
||||||
|
precision floor long before that.
|
||||||
|
\end{derivation}
|
||||||
|
|
||||||
|
\begin{limitation}
|
||||||
|
\textbf{TMJets on the full 10-state PKE is stuck at $\sim$10-second
|
||||||
|
horizons.} Prompt-neutron stiffness ($\Lambda = 10^{-4}$~\unit{\second}
|
||||||
|
vs.\ thermal $\sim 10$~\unit{\second}, ratio $10^5$) forces tiny
|
||||||
|
time steps. Step budget runs out, then numerical underflow
|
||||||
|
produces NaN in the precursor-decay arithmetic. Not a framework
|
||||||
|
bug; not a tuning issue; a fundamental property of reach on the
|
||||||
|
full PKE.
|
||||||
|
\end{limitation}
|
||||||
|
|
||||||
|
\subsection*{The singular-perturbation remedy (not done yet)}
|
||||||
|
|
||||||
|
\begin{decision}
|
||||||
|
For hours-long reach, the prompt neutron timescale must be eliminated
|
||||||
|
from the state dynamics. Standard move in reactor-kinetics reach
|
||||||
|
literature: set $\dot n \approx 0$ and solve for $n$ algebraically.
|
||||||
|
This gives the \emph{prompt-jump approximation} (sometimes written
|
||||||
|
as the \emph{reduced-order PKE}):
|
||||||
|
$$(\rho - \beta) n / \Lambda + \sum_i \lambda_i C_i = 0
|
||||||
|
\quad \Longrightarrow \quad
|
||||||
|
n \approx \frac{\Lambda \sum_i \lambda_i C_i}{\beta - \rho}$$
|
||||||
|
valid when $\rho < \beta$ (subcritical). $n$ becomes algebraic in
|
||||||
|
the remaining state; the dynamic state drops to 9-D, the stiffness
|
||||||
|
disappears, and TMJets can take steps on the thermal timescale
|
||||||
|
($\sim 1$~\unit{\second} per step is fine).
|
||||||
|
|
||||||
|
Tradeoff: we lose the prompt-supercritical regime, which is a
|
||||||
|
$\sim 50$~\unit{\microsecond} transient. For heatup over hours, that
|
||||||
|
transient is irrelevant to safety. Soundness is reduced (we no
|
||||||
|
longer capture prompt-dynamic trajectories), but the reduction
|
||||||
|
introduces a bounded error that can be characterized separately.
|
||||||
|
Deferred to a future session.
|
||||||
|
\end{decision}
|
||||||
|
|
||||||
|
\subsection*{Session result}
|
||||||
|
|
||||||
|
Artifacts:
|
||||||
|
\begin{itemize}
|
||||||
|
\item \texttt{reachability/predicates.json} restructured:
|
||||||
|
three-layer (deadbands, safety limits, mode invariants)
|
||||||
|
+ \texttt{mode\_boundaries} with per-mode entry/exit/time.
|
||||||
|
\item \texttt{reachability/load\_predicates.m} handles the new format
|
||||||
|
including multi-coefficient rows (for the rate halfspace).
|
||||||
|
\item \texttt{reachability/reach\_operation.m} and
|
||||||
|
\texttt{barrier\_lyapunov.m} now report per-halfspace margins
|
||||||
|
against \texttt{inv2\_holds}.
|
||||||
|
\item \texttt{reachability/barrier\_compare\_OL\_CL.m} — OL vs.\
|
||||||
|
CL Lyapunov barrier.
|
||||||
|
\item \texttt{reachability/WALKTHROUGH.md} — 550-line standalone
|
||||||
|
document.
|
||||||
|
\item \texttt{julia-port/scripts/reach\_heatup\_nonlinear.jl} and
|
||||||
|
\texttt{sim\_heatup.jl}. Nonlinear reach framework proven
|
||||||
|
to 10~\unit{\second}.
|
||||||
|
\item Six commits on \texttt{main}.
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
Key findings:
|
||||||
|
\begin{itemize}
|
||||||
|
\item Quadratic Lyapunov barrier fails structurally on this plant.
|
||||||
|
LQR improves it 20{,}000$\times$ but not enough. Need
|
||||||
|
polytopic or SOS barriers.
|
||||||
|
\item Heatup rate is a clean state halfspace (three-coefficient row).
|
||||||
|
Current controller peaks at 48.5~\unit{\celsius\per\hour} —
|
||||||
|
tight against a strict 28-spec interpretation.
|
||||||
|
\item Per-mode reach obligations split cleanly into equilibrium
|
||||||
|
(forever-invariance under disturbance) vs.\ transition
|
||||||
|
(reach-avoid with time budget, essentially zero disturbance).
|
||||||
|
\item Nonlinear reach framework functional in Julia at 10-second
|
||||||
|
horizons. Prompt-neutron stiffness is the wall; reduced-order
|
||||||
|
PKE is the known remedy.
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
\subsection*{Open at close}
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item Reduced-order PKE for long-horizon nonlinear reach.
|
||||||
|
\item Heatup reach to $T_{\max} = 5$~\unit{\hour}.
|
||||||
|
\item Scram reach (same reduction will apply).
|
||||||
|
\item Add polytopic or SOS barrier as alternative certificate.
|
||||||
|
\item Port MATLAB to Julia end-to-end; remove MATLAB dependency.
|
||||||
|
\item Build a FRET-adjacent UI for exploring the predicates
|
||||||
|
$\to$ halfspaces correspondence.
|
||||||
|
\item Lab journal to document all of the above (this is what got
|
||||||
|
done in the 2026-04-20 evening session — see next entry).
|
||||||
|
\end{itemize}
|
||||||
67
journal/journal.tex
Normal file
67
journal/journal.tex
Normal file
@ -0,0 +1,67 @@
|
|||||||
|
% journal.tex — top-level document that aggregates all dated entries.
|
||||||
|
%
|
||||||
|
% Build:
|
||||||
|
% cd journal && latexmk -pdf journal.tex
|
||||||
|
% or individual entry:
|
||||||
|
% cd journal && latexmk -pdf entries/2026-04-17-controllers-linear-reach.tex
|
||||||
|
|
||||||
|
\input{preamble.tex}
|
||||||
|
|
||||||
|
\title{HAHACS Lab Journal\\
|
||||||
|
\large PWR\_HYBRID\_3 preliminary example, invention log}
|
||||||
|
\author{Dane Sabo, with Claude (Hacker-Split)}
|
||||||
|
\date{Started \today}
|
||||||
|
|
||||||
|
\begin{document}
|
||||||
|
|
||||||
|
\maketitle
|
||||||
|
\tableofcontents
|
||||||
|
\newpage
|
||||||
|
|
||||||
|
\section*{How to read this journal}
|
||||||
|
|
||||||
|
Each section is a dated session. Sessions are written in two styles:
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item \textbf{Deep (A-style)}: full invention-log depth. Derivations
|
||||||
|
in math, code snippets with commentary, figures with long
|
||||||
|
captions, dead-ends documented, terminal output included where
|
||||||
|
it changes the story. A reader in 2030 should be able to
|
||||||
|
rebuild the work from this alone.
|
||||||
|
\item \textbf{Narrative (B-style)}: end-of-session notes with
|
||||||
|
pointers. Marked with \apass{some detail} callouts for
|
||||||
|
content that should be expanded in a later A-pass.
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
Callout boxes signal specific content types:
|
||||||
|
|
||||||
|
\begin{derivation}
|
||||||
|
Mathematical derivations — algebra, integrals, limits. Where they
|
||||||
|
matter to the safety claim, they live here in full.
|
||||||
|
\end{derivation}
|
||||||
|
|
||||||
|
\begin{decision}
|
||||||
|
Design decisions made during the session, with the rationale and the
|
||||||
|
alternatives considered.
|
||||||
|
\end{decision}
|
||||||
|
|
||||||
|
\begin{deadend}
|
||||||
|
Approaches that didn't work and why. These are as valuable as the
|
||||||
|
working paths — they keep the next explorer from repeating the mistake.
|
||||||
|
\end{deadend}
|
||||||
|
|
||||||
|
\begin{limitation}
|
||||||
|
Known-approximate or known-broken behavior. Soundness gaps live here.
|
||||||
|
Each limitation ties to a plan or an open question.
|
||||||
|
\end{limitation}
|
||||||
|
|
||||||
|
\newpage
|
||||||
|
|
||||||
|
% ---- Session entries, in chronological order -------------------------------
|
||||||
|
\input{entries/2026-04-17-controllers-linear-reach.tex}
|
||||||
|
\newpage
|
||||||
|
\input{entries/2026-04-20-predicates-boundaries-julia-nonlinear.tex}
|
||||||
|
\newpage
|
||||||
|
\input{entries/2026-04-20-evening-mega-session.tex}
|
||||||
|
|
||||||
|
\end{document}
|
||||||
176
journal/preamble.tex
Normal file
176
journal/preamble.tex
Normal file
@ -0,0 +1,176 @@
|
|||||||
|
% preamble.tex — shared LaTeX setup for the HAHACS lab journal.
|
||||||
|
%
|
||||||
|
% Intent: a reader in 2030 picks up this journal and can rebuild the
|
||||||
|
% work from scratch. This file defines the visual vocabulary — code
|
||||||
|
% listings, figures, callouts — that each dated entry uses.
|
||||||
|
|
||||||
|
\documentclass[11pt, letterpaper]{article}
|
||||||
|
|
||||||
|
% --- Page geometry ------------------------------------------------------------
|
||||||
|
\usepackage[margin=1in, headheight=15pt]{geometry}
|
||||||
|
\usepackage{parskip} % blank line between paragraphs; no indent
|
||||||
|
\usepackage{microtype}
|
||||||
|
|
||||||
|
% --- Fonts --------------------------------------------------------------------
|
||||||
|
\usepackage[T1]{fontenc}
|
||||||
|
\usepackage{lmodern}
|
||||||
|
\usepackage{inconsolata} % monospaced font for code
|
||||||
|
|
||||||
|
% --- Math ---------------------------------------------------------------------
|
||||||
|
\usepackage{amsmath, amssymb, amsthm, mathtools}
|
||||||
|
\usepackage{siunitx}
|
||||||
|
\sisetup{detect-all, per-mode=symbol}
|
||||||
|
|
||||||
|
% --- Graphics -----------------------------------------------------------------
|
||||||
|
\usepackage{graphicx}
|
||||||
|
\graphicspath{{./figures/}{../docs/figures/}}
|
||||||
|
\usepackage{caption}
|
||||||
|
\captionsetup{font=small, labelfont=bf}
|
||||||
|
|
||||||
|
% --- Code listings ------------------------------------------------------------
|
||||||
|
\usepackage{listings}
|
||||||
|
\usepackage{xcolor}
|
||||||
|
|
||||||
|
% Palette
|
||||||
|
\definecolor{code-bg}{HTML}{F7F7F2}
|
||||||
|
\definecolor{code-rule}{HTML}{D0D0C8}
|
||||||
|
\definecolor{code-comment}{HTML}{708070}
|
||||||
|
\definecolor{code-keyword}{HTML}{0050A0}
|
||||||
|
\definecolor{code-string}{HTML}{A04060}
|
||||||
|
\definecolor{code-number}{HTML}{806010}
|
||||||
|
|
||||||
|
\lstdefinelanguage{Julia}%
|
||||||
|
{morekeywords={abstract,break,case,catch,const,continue,do,else,elseif,%
|
||||||
|
end,export,false,for,function,immutable,import,importall,if,in,%
|
||||||
|
macro,module,otherwise,quote,return,struct,switch,true,try,type,%
|
||||||
|
typealias,using,while,mutable,global,local,let,primitive,where},%
|
||||||
|
sensitive=true,%
|
||||||
|
morecomment=[l]\#,%
|
||||||
|
morecomment=[n]{\#=}{=\#},%
|
||||||
|
morestring=[s]{"}{"},%
|
||||||
|
morestring=[m]{'}{'},%
|
||||||
|
}
|
||||||
|
|
||||||
|
\lstdefinestyle{labstyle}{
|
||||||
|
backgroundcolor=\color{code-bg},
|
||||||
|
basicstyle=\ttfamily\small,
|
||||||
|
breaklines=true,
|
||||||
|
breakatwhitespace=false,
|
||||||
|
captionpos=b,
|
||||||
|
commentstyle=\color{code-comment}\itshape,
|
||||||
|
frame=single,
|
||||||
|
framerule=0.5pt,
|
||||||
|
rulecolor=\color{code-rule},
|
||||||
|
keywordstyle=\color{code-keyword}\bfseries,
|
||||||
|
numberstyle=\tiny\color{code-comment},
|
||||||
|
showspaces=false,
|
||||||
|
showstringspaces=false,
|
||||||
|
showtabs=false,
|
||||||
|
stringstyle=\color{code-string},
|
||||||
|
tabsize=2,
|
||||||
|
numbers=left,
|
||||||
|
numbersep=6pt,
|
||||||
|
xleftmargin=14pt
|
||||||
|
}
|
||||||
|
\lstset{style=labstyle}
|
||||||
|
|
||||||
|
\lstdefinestyle{terminal}{
|
||||||
|
backgroundcolor=\color{black!90},
|
||||||
|
basicstyle=\ttfamily\small\color{green!80!white},
|
||||||
|
commentstyle=\color{white!60},
|
||||||
|
frame=none,
|
||||||
|
numbers=none,
|
||||||
|
breaklines=true,
|
||||||
|
xleftmargin=0pt
|
||||||
|
}
|
||||||
|
|
||||||
|
% --- Callout boxes ------------------------------------------------------------
|
||||||
|
\usepackage[most]{tcolorbox}
|
||||||
|
|
||||||
|
\newtcolorbox{limitation}[1][]{%
|
||||||
|
colback=red!5!white,
|
||||||
|
colframe=red!55!black,
|
||||||
|
title=\textbf{Limitation},
|
||||||
|
fonttitle=\bfseries,
|
||||||
|
boxrule=0.6pt,
|
||||||
|
arc=2pt,
|
||||||
|
left=6pt, right=6pt, top=4pt, bottom=4pt,
|
||||||
|
#1
|
||||||
|
}
|
||||||
|
|
||||||
|
\newtcolorbox{derivation}[1][]{%
|
||||||
|
colback=blue!3!white,
|
||||||
|
colframe=blue!40!black,
|
||||||
|
title=\textbf{Derivation},
|
||||||
|
fonttitle=\bfseries,
|
||||||
|
boxrule=0.6pt,
|
||||||
|
arc=2pt,
|
||||||
|
left=6pt, right=6pt, top=4pt, bottom=4pt,
|
||||||
|
#1
|
||||||
|
}
|
||||||
|
|
||||||
|
\newtcolorbox{deadend}[1][]{%
|
||||||
|
colback=gray!8!white,
|
||||||
|
colframe=gray!50!black,
|
||||||
|
title=\textbf{Dead end},
|
||||||
|
fonttitle=\bfseries,
|
||||||
|
boxrule=0.5pt,
|
||||||
|
arc=2pt,
|
||||||
|
borderline west={2pt}{0pt}{gray!40},
|
||||||
|
left=6pt, right=6pt, top=4pt, bottom=4pt,
|
||||||
|
#1
|
||||||
|
}
|
||||||
|
|
||||||
|
\newtcolorbox{decision}[1][]{%
|
||||||
|
colback=green!3!white,
|
||||||
|
colframe=green!35!black,
|
||||||
|
title=\textbf{Decision},
|
||||||
|
fonttitle=\bfseries,
|
||||||
|
boxrule=0.6pt,
|
||||||
|
arc=2pt,
|
||||||
|
left=6pt, right=6pt, top=4pt, bottom=4pt,
|
||||||
|
#1
|
||||||
|
}
|
||||||
|
|
||||||
|
% TODO-for-A-pass marker: highlighted so we can find them for later deep-pass.
|
||||||
|
\newcommand{\apass}[1]{%
|
||||||
|
{\fboxsep=2pt\colorbox{orange!20}{\textbf{[A-pass]}}\,\textit{#1}}
|
||||||
|
}
|
||||||
|
|
||||||
|
% --- Session header ----------------------------------------------------------
|
||||||
|
% Usage: \session{2026-04-17}{afternoon, ~5 hr}{summary line}
|
||||||
|
\newcommand{\session}[3]{%
|
||||||
|
\noindent\rule{\linewidth}{0.8pt}
|
||||||
|
\vspace{2pt}
|
||||||
|
\noindent\textbf{\Large Session: #1} \hfill \textit{#2}\par
|
||||||
|
\vspace{2pt}
|
||||||
|
\noindent\emph{#3}\par
|
||||||
|
\vspace{4pt}
|
||||||
|
\noindent\rule{\linewidth}{0.4pt}
|
||||||
|
\vspace{6pt}
|
||||||
|
}
|
||||||
|
|
||||||
|
% --- Code macros --------------------------------------------------------------
|
||||||
|
\newcommand{\juliafile}[2][]{\lstinputlisting[language=Julia,caption={#2},#1]{#2}}
|
||||||
|
\newcommand{\matlabfile}[2][]{\lstinputlisting[language=Matlab,caption={#2},#1]{#2}}
|
||||||
|
\newcommand{\pyfile}[2][]{\lstinputlisting[language=Python,caption={#2},#1]{#2}}
|
||||||
|
|
||||||
|
% --- Cross-reference helpers --------------------------------------------------
|
||||||
|
\usepackage[colorlinks=true, linkcolor=blue!50!black, citecolor=blue!40!black,
|
||||||
|
urlcolor=teal!60!black]{hyperref}
|
||||||
|
\usepackage{bookmark}
|
||||||
|
\usepackage{cleveref}
|
||||||
|
|
||||||
|
% --- Headers & footers --------------------------------------------------------
|
||||||
|
\usepackage{fancyhdr}
|
||||||
|
\pagestyle{fancy}
|
||||||
|
\fancyhf{}
|
||||||
|
\fancyhead[L]{HAHACS lab journal}
|
||||||
|
\fancyhead[R]{\leftmark}
|
||||||
|
\fancyfoot[C]{\thepage}
|
||||||
|
\renewcommand{\headrulewidth}{0.3pt}
|
||||||
|
|
||||||
|
% --- A small quiet easter egg for anyone who greps the preamble ---------------
|
||||||
|
% "Looks ordinary on the surface but is something else underneath."
|
||||||
|
% — the name behind Split. Keep the journal honest; under every summary
|
||||||
|
% there's a derivation; under every derivation there's a doubt.
|
||||||
82
julia-port/scripts/barrier_compare_OL_CL.jl
Normal file
82
julia-port/scripts/barrier_compare_OL_CL.jl
Normal file
@ -0,0 +1,82 @@
|
|||||||
|
#!/usr/bin/env julia
|
||||||
|
#
|
||||||
|
# barrier_compare_OL_CL.jl — head-to-head Lyapunov barrier bounds with
|
||||||
|
# and without LQR feedback. Julia port of reachability/barrier_compare_OL_CL.m.
|
||||||
|
#
|
||||||
|
# Expected: LQR improves every bound by ~20,000x, but bounds stay
|
||||||
|
# physically meaningless. Point is to show the ceiling is plant
|
||||||
|
# anisotropy, not controller tuning.
|
||||||
|
|
||||||
|
using Pkg
|
||||||
|
Pkg.activate(joinpath(@__DIR__, ".."))
|
||||||
|
|
||||||
|
using Printf
|
||||||
|
using LinearAlgebra
|
||||||
|
using MatrixEquations
|
||||||
|
using JSON
|
||||||
|
|
||||||
|
include(joinpath(@__DIR__, "..", "src", "pke_params.jl"))
|
||||||
|
include(joinpath(@__DIR__, "..", "src", "pke_th_rhs.jl"))
|
||||||
|
include(joinpath(@__DIR__, "..", "src", "pke_linearize.jl"))
|
||||||
|
include(joinpath(@__DIR__, "..", "src", "load_predicates.jl"))
|
||||||
|
|
||||||
|
plant = pke_params()
|
||||||
|
x_op = pke_initial_conditions(plant)
|
||||||
|
pred = load_predicates(plant)
|
||||||
|
|
||||||
|
A, B, B_w, _, _, _ = pke_linearize(plant)
|
||||||
|
|
||||||
|
# LQR gain (same weights as ctrl_operation_lqr).
|
||||||
|
Q_lqr = Diagonal([1.0, 1e-3, 1e-3, 1e-3, 1e-3, 1e-3, 1e-3, 1e-2, 1e2, 1.0])
|
||||||
|
R_lqr = 1e6 * ones(1, 1)
|
||||||
|
X_ric, _, _ = arec(A, reshape(B, :, 1), R_lqr, Matrix(Q_lqr))
|
||||||
|
K = (R_lqr \ reshape(B, 1, :)) * X_ric
|
||||||
|
A_cl = A - reshape(B, :, 1) * K
|
||||||
|
|
||||||
|
# Shared Qbar — best from the earlier sweep.
|
||||||
|
Qbar = Diagonal([1.0, 1e-4, 1e-4, 1e-4, 1e-4, 1e-4, 1e-4, 1.0, 1e2, 1.0])
|
||||||
|
w_bar = 0.15 * plant.P0
|
||||||
|
|
||||||
|
delta_entry = [0.01 * x_op[1];
|
||||||
|
0.001 .* abs.(x_op[2:7]);
|
||||||
|
0.1; 0.1; 0.1]
|
||||||
|
|
||||||
|
inv2 = pred.mode_invariants[:inv2_holds]
|
||||||
|
A_inv = inv2.A_poly
|
||||||
|
b_inv = inv2.b_poly
|
||||||
|
comps = inv2.components
|
||||||
|
b_dev = b_inv .- A_inv * x_op
|
||||||
|
|
||||||
|
function run_case(Acase)
|
||||||
|
P = lyapc(Matrix(Acase)', Matrix(Qbar))
|
||||||
|
Ph = Hermitian(sqrt(Hermitian(P)))
|
||||||
|
Phi = inv(Ph)
|
||||||
|
mu = minimum(eigvals(Symmetric(Phi * Matrix(Qbar) * Phi)))
|
||||||
|
g_bound = sqrt(B_w' * P * B_w)
|
||||||
|
c_inv = (2 * w_bar * g_bound / mu)^2
|
||||||
|
c_entry = delta_entry' * abs.(P) * delta_entry
|
||||||
|
gamma = max(c_inv, c_entry)
|
||||||
|
Pinv = inv(P)
|
||||||
|
maxima = [sqrt(gamma * (A_inv[k, :]' * Pinv * A_inv[k, :])) for k in 1:size(A_inv, 1)]
|
||||||
|
return gamma, maxima, eigvals(Acase)
|
||||||
|
end
|
||||||
|
|
||||||
|
gamma_OL, max_OL, eig_OL = run_case(A)
|
||||||
|
gamma_CL, max_CL, eig_CL = run_case(A_cl)
|
||||||
|
|
||||||
|
println("\n=== Open-loop vs LQR closed-loop Lyapunov barrier ===")
|
||||||
|
@printf " max Re(eig) OL = %.3e CL = %.3e\n" maximum(real.(eig_OL)) maximum(real.(eig_CL))
|
||||||
|
@printf " min Re(eig) OL = %.3e CL = %.3e\n" minimum(real.(eig_OL)) minimum(real.(eig_CL))
|
||||||
|
@printf " gamma OL = %.3e CL = %.3e (ratio CL/OL = %.3g)\n" gamma_OL gamma_CL gamma_CL/gamma_OL
|
||||||
|
|
||||||
|
println("\n Per-halfspace max |a' dx| on gamma-ellipsoid:")
|
||||||
|
@printf " %-22s %-10s %-14s %-14s %-14s %-10s\n" "halfspace" "headroom" "open-loop" "closed-loop" "CL - OL" "ratio"
|
||||||
|
for k in 1:size(A_inv, 1)
|
||||||
|
ratio = max_CL[k] / max_OL[k]
|
||||||
|
@printf " %-22s %10.3f %14.3f %14.3f %+14.3f %10.3gx\n" comps[k] b_dev[k] max_OL[k] max_CL[k] (max_CL[k] - max_OL[k]) ratio
|
||||||
|
end
|
||||||
|
|
||||||
|
println("\n Interpretation:")
|
||||||
|
println(" - CL < OL on a row => LQR tightens that halfspace.")
|
||||||
|
println(" - CL ~= OL => LQR does not help that direction at all.")
|
||||||
|
println(" - CL > OL => LQR made that direction WORSE for the barrier.")
|
||||||
118
julia-port/scripts/barrier_lyapunov.jl
Normal file
118
julia-port/scripts/barrier_lyapunov.jl
Normal file
@ -0,0 +1,118 @@
|
|||||||
|
#!/usr/bin/env julia
|
||||||
|
#
|
||||||
|
# barrier_lyapunov.jl — Lyapunov-ellipsoid barrier cert.
|
||||||
|
# Julia port of reachability/barrier_lyapunov.m.
|
||||||
|
#
|
||||||
|
# Sweeps Qbar(T_c) weight to find the tightest quadratic barrier;
|
||||||
|
# reports per-halfspace margins against inv2_holds.
|
||||||
|
#
|
||||||
|
# Expected result: every halfspace exceeded by orders of magnitude.
|
||||||
|
# This is the structural anisotropy limitation, not a code bug.
|
||||||
|
|
||||||
|
using Pkg
|
||||||
|
Pkg.activate(joinpath(@__DIR__, ".."))
|
||||||
|
|
||||||
|
using Printf
|
||||||
|
using LinearAlgebra
|
||||||
|
using MatrixEquations
|
||||||
|
using JSON
|
||||||
|
|
||||||
|
include(joinpath(@__DIR__, "..", "src", "pke_params.jl"))
|
||||||
|
include(joinpath(@__DIR__, "..", "src", "pke_th_rhs.jl"))
|
||||||
|
include(joinpath(@__DIR__, "..", "src", "pke_linearize.jl"))
|
||||||
|
include(joinpath(@__DIR__, "..", "src", "load_predicates.jl"))
|
||||||
|
|
||||||
|
plant = pke_params()
|
||||||
|
x_op = pke_initial_conditions(plant)
|
||||||
|
pred = load_predicates(plant)
|
||||||
|
|
||||||
|
# --- Closed-loop ---
|
||||||
|
A, B, B_w, _, _, _ = pke_linearize(plant)
|
||||||
|
Q_lqr = Diagonal([1.0, 1e-3, 1e-3, 1e-3, 1e-3, 1e-3, 1e-3, 1e-2, 1e2, 1.0])
|
||||||
|
R_lqr = 1e6 * ones(1, 1)
|
||||||
|
X_ric, _, _ = arec(A, reshape(B, :, 1), R_lqr, Matrix(Q_lqr))
|
||||||
|
K = (R_lqr \ reshape(B, 1, :)) * X_ric
|
||||||
|
A_cl = A - reshape(B, :, 1) * K
|
||||||
|
|
||||||
|
# --- Lyapunov ---
|
||||||
|
# lyapc solves A' P + P A + Q = 0 for P. Julia's MatrixEquations uses
|
||||||
|
# lyapc(A', Q) to solve A' P + P A = -Q (note: sign conventions vary).
|
||||||
|
# Check: for a Hurwitz A, lyapc(A, Q) with Q > 0 returns P > 0.
|
||||||
|
Qbar = Diagonal([1.0, 1e-4, 1e-4, 1e-4, 1e-4, 1e-4, 1e-4, 1.0, 1e2, 1.0])
|
||||||
|
P = lyapc(Matrix(A_cl)', Matrix(Qbar))
|
||||||
|
@assert all(eigvals(P) .> 0) "P not positive definite"
|
||||||
|
|
||||||
|
# --- Disturbance bound ---
|
||||||
|
w_bar = 0.15 * plant.P0
|
||||||
|
|
||||||
|
# Invariant level via the derivation in the MATLAB file.
|
||||||
|
P_half = Hermitian(sqrt(Hermitian(P)))
|
||||||
|
P_half_inv = inv(P_half)
|
||||||
|
mu = minimum(eigvals(Symmetric(P_half_inv * Matrix(Qbar) * P_half_inv)))
|
||||||
|
g_bound = sqrt(B_w' * P * B_w)
|
||||||
|
c_inv = (2 * w_bar * g_bound / mu)^2
|
||||||
|
|
||||||
|
delta_entry = [0.01 * x_op[1];
|
||||||
|
0.001 .* abs.(x_op[2:7]);
|
||||||
|
0.1; 0.1; 0.1]
|
||||||
|
c_entry = delta_entry' * abs.(P) * delta_entry
|
||||||
|
|
||||||
|
gamma = max(c_inv, c_entry)
|
||||||
|
|
||||||
|
println("\n=== Lyapunov barrier certificate ===")
|
||||||
|
@printf " lambda_min(P) = %.3e\n" minimum(eigvals(P))
|
||||||
|
@printf " lambda_max(P) = %.3e\n" maximum(eigvals(P))
|
||||||
|
@printf " sqrt(B_w' P B_w) = %.3e\n" g_bound
|
||||||
|
@printf " mu (Qbar eig on P-metric)= %.3e\n" mu
|
||||||
|
@printf " w_bar (15%% P0) = %.3e W\n" w_bar
|
||||||
|
@printf " c_inv (invariant level) = %.3e\n" c_inv
|
||||||
|
@printf " c_entry (from X_entry) = %.3e\n" c_entry
|
||||||
|
@printf " gamma = %.3e\n" gamma
|
||||||
|
|
||||||
|
# --- inv2_holds per-halfspace barrier check ---
|
||||||
|
inv2 = pred.mode_invariants[:inv2_holds]
|
||||||
|
A_inv = inv2.A_poly
|
||||||
|
b_inv = inv2.b_poly
|
||||||
|
comps = inv2.components
|
||||||
|
b_dev = b_inv .- A_inv * x_op
|
||||||
|
|
||||||
|
Pinv = inv(P)
|
||||||
|
println("\n=== Lyapunov barrier vs inv2_holds halfspaces ===")
|
||||||
|
for k in 1:size(A_inv, 1)
|
||||||
|
a = A_inv[k, :]
|
||||||
|
max_adx = sqrt(gamma * (a' * Pinv * a))
|
||||||
|
margin = b_dev[k] - max_adx
|
||||||
|
status = margin < 0 ? "*** TOO LOOSE ***" : "OK"
|
||||||
|
@printf " [%-22s] headroom = %8.3f | max a'dx = %8.3f | margin = %+8.3f %s\n" comps[k] b_dev[k] max_adx margin status
|
||||||
|
end
|
||||||
|
|
||||||
|
# --- Qbar(T_c) sweep for tightness ---
|
||||||
|
println("\n=== Qbar(T_c) weight sweep ===")
|
||||||
|
e9 = zeros(10); e9[9] = 1.0
|
||||||
|
weights = [1e1, 1e2, 1e3, 1e4, 1e5, 1e6]
|
||||||
|
let best_dTc = Inf, best_w = NaN, best_gamma = NaN, best_P = nothing
|
||||||
|
for wTc in weights
|
||||||
|
Qbs = copy(Matrix(Qbar))
|
||||||
|
Qbs[9, 9] = wTc
|
||||||
|
Ps = try
|
||||||
|
lyapc(Matrix(A_cl)', Qbs)
|
||||||
|
catch
|
||||||
|
continue
|
||||||
|
end
|
||||||
|
any(eigvals(Ps) .<= 0) && continue
|
||||||
|
Phs = Hermitian(sqrt(Hermitian(Ps)))
|
||||||
|
Phi = inv(Phs)
|
||||||
|
mu_s = minimum(eigvals(Symmetric(Phi * Qbs * Phi)))
|
||||||
|
g_s = sqrt(B_w' * Ps * B_w)
|
||||||
|
ci_s = (2 * w_bar * g_s / mu_s)^2
|
||||||
|
ce_s = delta_entry' * abs.(Ps) * delta_entry
|
||||||
|
gs = max(ci_s, ce_s)
|
||||||
|
Pinv_s = inv(Ps)
|
||||||
|
dTc = sqrt(gs * (e9' * Pinv_s * e9))
|
||||||
|
@printf " Qbar(9,9) = %.0e -> gamma = %.3e, max|dT_c| = %7.3f K\n" wTc gs dTc
|
||||||
|
if dTc < best_dTc
|
||||||
|
best_dTc = dTc; best_w = wTc; best_gamma = gs; best_P = Ps
|
||||||
|
end
|
||||||
|
end
|
||||||
|
@printf " Best: Qbar(9,9) = %.0e -> max|dT_c| = %.3f K\n" best_w best_dTc
|
||||||
|
end
|
||||||
132
julia-port/scripts/main_mode_sweep.jl
Normal file
132
julia-port/scripts/main_mode_sweep.jl
Normal file
@ -0,0 +1,132 @@
|
|||||||
|
#!/usr/bin/env julia
|
||||||
|
#
|
||||||
|
# main_mode_sweep.jl — run every DRC-mode controller back-to-back.
|
||||||
|
#
|
||||||
|
# Julia equivalent of plant-model/main_mode_sweep.m. Produces the
|
||||||
|
# same four-panel plots, saves to docs/figures/ with `mode_sweep_*.png`
|
||||||
|
# names (overwrites MATLAB outputs — the Julia versions take over).
|
||||||
|
|
||||||
|
using Pkg
|
||||||
|
Pkg.activate(joinpath(@__DIR__, ".."))
|
||||||
|
|
||||||
|
using Printf
|
||||||
|
using LinearAlgebra
|
||||||
|
using OrdinaryDiffEq
|
||||||
|
using Plots
|
||||||
|
using MatrixEquations
|
||||||
|
using JSON
|
||||||
|
|
||||||
|
include(joinpath(@__DIR__, "..", "src", "pke_params.jl"))
|
||||||
|
include(joinpath(@__DIR__, "..", "src", "pke_th_rhs.jl"))
|
||||||
|
include(joinpath(@__DIR__, "..", "src", "pke_linearize.jl"))
|
||||||
|
include(joinpath(@__DIR__, "..", "src", "pke_solver.jl"))
|
||||||
|
include(joinpath(@__DIR__, "..", "src", "plot_pke_results.jl"))
|
||||||
|
include(joinpath(@__DIR__, "..", "controllers", "controllers.jl"))
|
||||||
|
|
||||||
|
# --- Plant + predicates ---
|
||||||
|
plant = pke_params()
|
||||||
|
|
||||||
|
# Load T_standby from predicates.json for the hot-standby IC + heatup ref.
|
||||||
|
pred_path = joinpath(@__DIR__, "..", "..", "reachability", "predicates.json")
|
||||||
|
pred_raw = JSON.parsefile(pred_path)
|
||||||
|
T_standby = plant.T_c0 + pred_raw["derived"]["T_standby_offset_C"]
|
||||||
|
|
||||||
|
# --- ICs ---
|
||||||
|
x_op = pke_initial_conditions(plant)
|
||||||
|
|
||||||
|
# Hot-standby IC: everything at T_standby, trace n.
|
||||||
|
n_shut = 1e-6
|
||||||
|
C_shut = (plant.beta_i ./ (plant.lambda_i .* plant.Lambda)) .* n_shut
|
||||||
|
x_shut = [n_shut; C_shut; T_standby; T_standby; T_standby]
|
||||||
|
|
||||||
|
# Heatup IC: already critical at 0.1% power, thermally matched.
|
||||||
|
n_heat = 1e-3
|
||||||
|
C_heat = (plant.beta_i ./ (plant.lambda_i .* plant.Lambda)) .* n_heat
|
||||||
|
x_heat = [n_heat; C_heat; T_standby; T_standby; T_standby]
|
||||||
|
|
||||||
|
# --- LQR gain (needed by ctrl_operation_lqr_factory) ---
|
||||||
|
A, B, B_w, _, _, _ = pke_linearize(plant)
|
||||||
|
Q_lqr = Diagonal([1.0, 1e-3, 1e-3, 1e-3, 1e-3, 1e-3, 1e-3, 1e-2, 1e2, 1.0])
|
||||||
|
R_lqr = 1e6 * ones(1, 1)
|
||||||
|
ctrl_op_lqr, K_lqr, _ = ctrl_operation_lqr_factory(
|
||||||
|
plant, A, B; Q_lqr=Matrix(Q_lqr), R_lqr=R_lqr)
|
||||||
|
|
||||||
|
# --- Figure output directory ---
|
||||||
|
figdir = joinpath(@__DIR__, "..", "..", "docs", "figures")
|
||||||
|
isdir(figdir) || mkpath(figdir)
|
||||||
|
|
||||||
|
# --- Run each mode ---
|
||||||
|
|
||||||
|
println("\n===== Mode 1: ctrl_shutdown =====")
|
||||||
|
Q_shut = t -> 0.0
|
||||||
|
t1, X1, U1 = pke_solver(plant, Q_shut, ctrl_shutdown, nothing, (0.0, 600.0); x0=x_shut)
|
||||||
|
|
||||||
|
println("\n===== Mode 2: ctrl_heatup =====")
|
||||||
|
ref_heat = (; T_start=T_standby, T_target=plant.T_c0, ramp_rate=28/3600)
|
||||||
|
Q_heat = t -> 0.0
|
||||||
|
t2, X2, U2 = pke_solver(plant, Q_heat, ctrl_heatup, ref_heat, (0.0, 5400.0); x0=x_heat)
|
||||||
|
|
||||||
|
println("\n===== Mode 3a: ctrl_operation (P) =====")
|
||||||
|
Q_op = t -> plant.P0 * (1.0 - 0.2 * (t >= 30))
|
||||||
|
ref_op = (; T_avg=plant.T_c0)
|
||||||
|
t3, X3, U3 = pke_solver(plant, Q_op, ctrl_operation, ref_op, (0.0, 600.0))
|
||||||
|
|
||||||
|
println("\n===== Mode 3b: ctrl_operation_lqr =====")
|
||||||
|
# LQR closure ignores ref; pass nothing.
|
||||||
|
t3b, X3b, U3b = pke_solver(plant, Q_op, ctrl_op_lqr, nothing, (0.0, 600.0))
|
||||||
|
|
||||||
|
println("\n===== Mode 4: ctrl_scram =====")
|
||||||
|
Q_scr = t -> plant.P0 * max(0.03, 1 - max(0, t - 10) / 10)
|
||||||
|
t4, X4, U4 = pke_solver(plant, Q_scr, ctrl_scram, nothing, (0.0, 600.0))
|
||||||
|
|
||||||
|
# --- Per-mode 4-panel plots ---
|
||||||
|
plot_pke_results(t1, X1, U1, plant, Q_shut;
|
||||||
|
title="ctrl_shutdown (cold IC)",
|
||||||
|
savepath=joinpath(figdir, "mode_sweep_1_shutdown.png"))
|
||||||
|
plot_pke_results(t2, X2, U2, plant, Q_heat;
|
||||||
|
title="ctrl_heatup (ramp T_avg)",
|
||||||
|
savepath=joinpath(figdir, "mode_sweep_2_heatup.png"))
|
||||||
|
plot_pke_results(t3, X3, U3, plant, Q_op;
|
||||||
|
title="ctrl_operation (P on T_avg)",
|
||||||
|
savepath=joinpath(figdir, "mode_sweep_3_operation.png"))
|
||||||
|
plot_pke_results(t3b, X3b, U3b, plant, Q_op;
|
||||||
|
title="ctrl_operation_lqr",
|
||||||
|
savepath=joinpath(figdir, "mode_sweep_3b_operation_lqr.png"))
|
||||||
|
plot_pke_results(t4, X4, U4, plant, Q_scr;
|
||||||
|
title="ctrl_scram",
|
||||||
|
savepath=joinpath(figdir, "mode_sweep_4_scram.png"))
|
||||||
|
|
||||||
|
# --- Heatup ramp-tracking figure ---
|
||||||
|
CtoF(T) = T*9/5 + 32
|
||||||
|
T_ref_trace = [min(T_standby + (28/3600)*ti, plant.T_c0) for ti in t2]
|
||||||
|
fig_heat = plot(t2 ./ 60, CtoF.(T_ref_trace), lw=1.3, ls=:dash, color=:black,
|
||||||
|
label="T_ref (28 C/hr ramp)",
|
||||||
|
xlabel="Time [min]", ylabel="T_avg [F]",
|
||||||
|
title="ctrl_heatup: reference tracking", size=(1000, 400))
|
||||||
|
plot!(fig_heat, t2 ./ 60, CtoF.(X2[:, 9]), lw=2, color=:red,
|
||||||
|
label="T_avg (plant)")
|
||||||
|
savefig(fig_heat, joinpath(figdir, "mode_sweep_heatup_tracking.png"))
|
||||||
|
|
||||||
|
# --- P vs LQR head-to-head ---
|
||||||
|
fig_ctrl = plot(t3, CtoF.(X3[:, 9]), lw=2, color=:blue, label="ctrl_operation (P)",
|
||||||
|
xlabel="Time [s]", ylabel="T_avg [F]",
|
||||||
|
title="Operation mode: P vs LQR under 100% -> 80% Q_sg step",
|
||||||
|
size=(1000, 400))
|
||||||
|
plot!(fig_ctrl, t3b, CtoF.(X3b[:, 9]), lw=2, color=:red, label="ctrl_operation_lqr")
|
||||||
|
hline!(fig_ctrl, [CtoF(plant.T_c0)], ls=:dash, color=:black, label="setpoint")
|
||||||
|
savefig(fig_ctrl, joinpath(figdir, "mode_sweep_op_P_vs_LQR.png"))
|
||||||
|
|
||||||
|
# --- Power overview ---
|
||||||
|
p_shut = plot(t1, max.(X1[:,1], 1e-20), yaxis=:log, title="Shutdown",
|
||||||
|
xlabel="t [s]", ylabel="n (log)", legend=false, lw=1.5)
|
||||||
|
p_heat = plot(t2 ./ 60, X2[:,1], title="Heatup", xlabel="t [min]", ylabel="n",
|
||||||
|
legend=false, lw=1.5)
|
||||||
|
p_op = plot(t3, X3[:,1], title="Operation", xlabel="t [s]", ylabel="n",
|
||||||
|
legend=false, lw=1.5)
|
||||||
|
p_scr = plot(t4, max.(X4[:,1], 1e-20), yaxis=:log, title="Scram",
|
||||||
|
xlabel="t [s]", ylabel="n (log)", legend=false, lw=1.5)
|
||||||
|
fig_ov = plot(p_shut, p_heat, p_op, p_scr, layout=(2,2), size=(1100, 650),
|
||||||
|
plot_title="Normalized reactor power n(t) across DRC modes")
|
||||||
|
savefig(fig_ov, joinpath(figdir, "mode_sweep_power_overview.png"))
|
||||||
|
|
||||||
|
println("\n=== Figures written to $figdir ===")
|
||||||
@ -1,90 +1,150 @@
|
|||||||
#!/usr/bin/env julia
|
#!/usr/bin/env julia
|
||||||
#
|
#
|
||||||
# reach_operation.jl — operation-mode linear reach in Julia. **STUB**.
|
# reach_operation.jl — Julia port of reachability/reach_operation.m.
|
||||||
#
|
#
|
||||||
# Attempt to reproduce reachability/reach_operation.m using
|
# Linear reach tube for operation-mode LQR closed-loop. Per-halfspace
|
||||||
# ReachabilityAnalysis.jl.
|
# margin check against inv2_holds (from predicates.json, single source
|
||||||
|
# of truth). Figures saved to docs/figures/.
|
||||||
#
|
#
|
||||||
# STATUS: does NOT yet produce a useful result. The closed-loop system
|
# Soundness caveat: this reach tube is a sound over-approximation of the
|
||||||
# is strongly stiff (eigvals spanning 0.012 to 65 /s) and has states
|
# LINEARIZED closed-loop system's reach set, not of the nonlinear plant.
|
||||||
# with magnitudes differing by ~10 orders of magnitude (precursors C_i
|
# Linear-reach results here are an approximate-but-useful gut check.
|
||||||
# ~ 1e5 due to 1/Lambda, temperatures ~ 300). LGG09 / GLGM06 / BFFPSV18
|
|
||||||
# all blow up numerically over the 600s horizon, returning envelopes
|
|
||||||
# ranging from 2757 K to 1e37 K instead of the known-tight 0.03 K.
|
|
||||||
#
|
|
||||||
# Likely fix: diagonal state scaling S such that A_cl_scaled = S A_cl S^{-1}
|
|
||||||
# has O(1) entries, run reach in scaled coordinates, then invert S.
|
|
||||||
# Also worth trying: ASB07 (adaptive step) or the Taylor-model schemes.
|
|
||||||
# Left as future work — the hand-rolled MATLAB zonotope in
|
|
||||||
# reachability/reach_linear.m gives a valid result today, so the Julia
|
|
||||||
# port is priority-B.
|
|
||||||
|
|
||||||
using Pkg
|
using Pkg
|
||||||
Pkg.activate(joinpath(@__DIR__, ".."))
|
Pkg.activate(joinpath(@__DIR__, ".."))
|
||||||
|
|
||||||
|
using Printf
|
||||||
using LinearAlgebra
|
using LinearAlgebra
|
||||||
using MatrixEquations
|
using MatrixEquations
|
||||||
using ReachabilityAnalysis, LazySets
|
using Plots
|
||||||
|
using JSON
|
||||||
|
using MAT
|
||||||
|
|
||||||
include(joinpath(@__DIR__, "..", "src", "pke_params.jl"))
|
include(joinpath(@__DIR__, "..", "src", "pke_params.jl"))
|
||||||
include(joinpath(@__DIR__, "..", "src", "pke_th_rhs.jl"))
|
include(joinpath(@__DIR__, "..", "src", "pke_th_rhs.jl"))
|
||||||
include(joinpath(@__DIR__, "..", "src", "pke_linearize.jl"))
|
include(joinpath(@__DIR__, "..", "src", "pke_linearize.jl"))
|
||||||
|
include(joinpath(@__DIR__, "..", "src", "load_predicates.jl"))
|
||||||
|
include(joinpath(@__DIR__, "..", "src", "reach_linear.jl"))
|
||||||
|
|
||||||
plant = pke_params()
|
plant = pke_params()
|
||||||
x_op = pke_initial_conditions(plant)
|
x_op = pke_initial_conditions(plant)
|
||||||
|
pred = load_predicates(plant)
|
||||||
|
|
||||||
# --- Linearize around the operating point ---
|
# --- Closed-loop linearization + LQR gain ---
|
||||||
A, B, B_w, _, _, _ = pke_linearize(plant)
|
A, B, B_w, _, _, _ = pke_linearize(plant)
|
||||||
|
|
||||||
# --- LQR gain (same Q, R as the MATLAB side) ---
|
|
||||||
Q_lqr = Diagonal([1.0, 1e-3, 1e-3, 1e-3, 1e-3, 1e-3, 1e-3, 1e-2, 1e2, 1.0])
|
Q_lqr = Diagonal([1.0, 1e-3, 1e-3, 1e-3, 1e-3, 1e-3, 1e-3, 1e-2, 1e2, 1.0])
|
||||||
R_lqr = 1e6 * ones(1, 1)
|
R_lqr = 1e6 * ones(1, 1)
|
||||||
X_ric, _, _ = arec(A, reshape(B, :, 1), R_lqr, Matrix(Q_lqr))
|
X_ric, _, _ = arec(A, reshape(B, :, 1), R_lqr, Matrix(Q_lqr))
|
||||||
K = (R_lqr \ reshape(B, 1, :) * X_ric) # 1 x 10
|
K = (R_lqr \ reshape(B, 1, :)) * X_ric
|
||||||
A_cl = A - reshape(B, :, 1) * K
|
A_cl = A - reshape(B, :, 1) * K
|
||||||
|
|
||||||
println("\n=== Julia closed-loop spectrum ===")
|
eigs_cl = eigvals(A_cl)
|
||||||
ev = eigvals(A_cl)
|
println("\n=== Closed-loop spectrum (A - BK) ===")
|
||||||
println(" max Re(eig) = ", round(maximum(real.(ev)); sigdigits=4))
|
@printf " max Re(eig) = %.3e\n" maximum(real.(eigs_cl))
|
||||||
println(" min Re(eig) = ", round(minimum(real.(ev)); sigdigits=4))
|
@printf " min Re(eig) = %.3e\n" minimum(real.(eigs_cl))
|
||||||
|
@assert all(real.(eigs_cl) .< -1e-8) "A_cl not Hurwitz"
|
||||||
|
|
||||||
# --- Reach-set problem: dx/dt = A_cl dx + B_w w, dx(0) ∈ X0, w ∈ W ---
|
# --- Entry box ---
|
||||||
delta_entry = [0.01 * x_op[1];
|
delta_entry = [0.01 * x_op[1];
|
||||||
0.001 .* abs.(x_op[2:7]);
|
0.001 .* abs.(x_op[2:7]);
|
||||||
0.1; 0.1; 0.1]
|
0.1; 0.1; 0.1]
|
||||||
X0 = Hyperrectangle(zeros(10), delta_entry)
|
|
||||||
|
|
||||||
|
# --- Disturbance envelope ---
|
||||||
Q_nom = plant.P0
|
Q_nom = plant.P0
|
||||||
w_lo, w_hi = -0.15 * plant.P0, 0.0
|
Q_min = 0.85 * plant.P0
|
||||||
W = Interval(w_lo, w_hi)
|
Q_max = 1.00 * plant.P0
|
||||||
B_w_col = reshape(B_w, :, 1)
|
dQ_lo = Q_min - Q_nom
|
||||||
|
dQ_hi = Q_max - Q_nom
|
||||||
|
|
||||||
# Encode the bounded disturbance as a bounded input:
|
# --- Reach in deviation coordinates ---
|
||||||
# dx/dt = A_cl x + B_w u, u ∈ W.
|
tspan = (0.0, 600.0)
|
||||||
# Direct constructor since the @system macro's dialect is package-version sensitive.
|
dt = 0.5
|
||||||
sys = ConstrainedLinearControlContinuousSystem(A_cl, B_w_col, Universe(10), W)
|
T, R_lo, R_hi, Cnom = reach_linear(A_cl, B_w, zeros(10), delta_entry,
|
||||||
prob = InitialValueProblem(sys, X0)
|
dQ_lo, dQ_hi, tspan, dt)
|
||||||
sol = solve(prob; T=600.0, alg=GLGM06(; δ=0.5, max_order=20))
|
|
||||||
|
|
||||||
# Extract T_c envelope via support function queries.
|
# Translate back to absolute coordinates.
|
||||||
flow = flowpipe(sol)
|
X_lo = R_lo .+ x_op
|
||||||
e9 = zeros(10); e9[9] = 1.0
|
X_hi = R_hi .+ x_op
|
||||||
T_c_hi = [ρ(e9, set(R)) for R in flow]
|
X_nom = Cnom .+ x_op
|
||||||
T_c_lo = [-ρ(-e9, set(R)) for R in flow]
|
|
||||||
|
|
||||||
println("\n=== Operation-mode reach envelope on T_c (deviation from T_c0) ===")
|
# --- Safety check: inv2_holds halfspace-by-halfspace ---
|
||||||
println(" min dT_c = ", round(minimum(T_c_lo); digits=4), " K")
|
inv2 = pred.mode_invariants[:inv2_holds]
|
||||||
println(" max dT_c = ", round(maximum(T_c_hi); digits=4), " K")
|
A_inv = inv2.A_poly
|
||||||
println(" safety band |dT_c| <= 5.0 K")
|
b_inv = inv2.b_poly
|
||||||
if maximum(abs.([T_c_lo; T_c_hi])) <= 5.0
|
comps = inv2.components
|
||||||
println(" OK: Julia reach set inside safety band.")
|
|
||||||
else
|
println("\n=== Operation-mode reach vs inv2_holds safety limits ===")
|
||||||
println(" *** VIOLATION ***")
|
for k in 1:size(A_inv, 1)
|
||||||
|
a = A_inv[k, :]
|
||||||
|
# Envelope max of a'*x: take Xhi where a>0, Xlo where a<0.
|
||||||
|
env_for_max = (a .> 0) .* X_hi .+ (a .< 0) .* X_lo
|
||||||
|
# Small correction: zero coeffs have no preference (both values equal).
|
||||||
|
zero_mask = (a .== 0)
|
||||||
|
env_for_max[zero_mask, :] .= X_hi[zero_mask, :]
|
||||||
|
max_ax = maximum(a' * env_for_max)
|
||||||
|
margin = b_inv[k] - max_ax
|
||||||
|
status = margin < 0 ? "*** VIOLATED ***" : "OK"
|
||||||
|
@printf " [%-22s] a'x <= %8.3f | max a'x = %8.3f | margin = %+8.3f %s\n" comps[k] b_inv[k] max_ax margin status
|
||||||
end
|
end
|
||||||
|
|
||||||
# Save the envelope for later comparison
|
# --- Per-state reach-set width diagnostic ---
|
||||||
using MAT
|
state_names = ["n","C1","C2","C3","C4","C5","C6","T_f","T_c","T_cold"]
|
||||||
matwrite(joinpath(@__DIR__, "..", "..", "reachability", "julia_reach_operation.mat"),
|
println("\n=== Reach-set width at t=0 vs t=T_final ===")
|
||||||
Dict("T_c_lo" => T_c_lo, "T_c_hi" => T_c_hi,
|
@printf " %-7s %-14s %-14s %-8s\n" "state" "init halfwidth" "final halfwidth" "ratio"
|
||||||
"A_cl" => A_cl, "K" => Matrix(K), "delta_entry" => delta_entry))
|
for i in 1:10
|
||||||
println("\nSaved envelope to reachability/julia_reach_operation.mat")
|
hi = 0.5 * (R_hi[i, 1] - R_lo[i, 1])
|
||||||
|
hf = 0.5 * (R_hi[i, end] - R_lo[i, end])
|
||||||
|
@printf " %-7s %.4e %.4e %.2f\n" state_names[i] hi hf hf/max(hi, eps())
|
||||||
|
end
|
||||||
|
|
||||||
|
# --- Plots: T_c reach tube, two views ---
|
||||||
|
CtoF(T) = T * 9/5 + 32
|
||||||
|
delta_safe_Tc = pred.constants.t_avg_in_range_halfwidth_C
|
||||||
|
figdir = joinpath(@__DIR__, "..", "..", "docs", "figures")
|
||||||
|
isdir(figdir) || mkpath(figdir)
|
||||||
|
|
||||||
|
p_safety = plot(T, CtoF.(X_nom[9, :]), lw=1.2, color=:red,
|
||||||
|
label="nominal",
|
||||||
|
xlabel="Time [s]", ylabel="T_avg [F]",
|
||||||
|
title="Safety-band view")
|
||||||
|
plot!(p_safety, T, CtoF.(X_hi[9, :]), fillrange=CtoF.(X_lo[9, :]),
|
||||||
|
fillalpha=0.3, color=:red, linealpha=0.0,
|
||||||
|
label="reach tube")
|
||||||
|
hline!(p_safety, [CtoF(plant.T_c0 + delta_safe_Tc),
|
||||||
|
CtoF(plant.T_c0 - delta_safe_Tc)],
|
||||||
|
ls=:dash, color=:black, label="safety +/- $(round(delta_safe_Tc; digits=2)) C")
|
||||||
|
|
||||||
|
dev_lo = X_lo[9, :] .- plant.T_c0
|
||||||
|
dev_hi = X_hi[9, :] .- plant.T_c0
|
||||||
|
max_dev = maximum(abs, [dev_lo; dev_hi])
|
||||||
|
p_zoom = plot(T, X_nom[9, :] .- plant.T_c0, lw=1.2, color=:red,
|
||||||
|
label="nominal",
|
||||||
|
xlabel="Time [s]", ylabel="T_avg - T_c0 [K]",
|
||||||
|
title=@sprintf("Zoomed: max |dT_c| = %.3e K", max_dev))
|
||||||
|
plot!(p_zoom, T, dev_hi, fillrange=dev_lo, fillalpha=0.3,
|
||||||
|
color=:red, linealpha=0.0, label="reach tube")
|
||||||
|
hline!(p_zoom, [0.0], ls=:dot, color=:black, label=nothing)
|
||||||
|
|
||||||
|
fig = plot(p_safety, p_zoom, layout=(1, 2), size=(1400, 500),
|
||||||
|
plot_title=@sprintf("Operation-mode reach tube, LQR, Q_sg in [%.0f%%, %.0f%%] P0",
|
||||||
|
100*Q_min/Q_nom, 100*Q_max/Q_nom))
|
||||||
|
savefig(fig, joinpath(figdir, "reach_operation_Tc.png"))
|
||||||
|
|
||||||
|
# n tube
|
||||||
|
fig_n = plot(T, X_nom[1, :], lw=1.2, color=:blue, label="nominal",
|
||||||
|
xlabel="Time [s]", ylabel="n",
|
||||||
|
title="Operation mode reach tube on normalized power")
|
||||||
|
plot!(fig_n, T, X_hi[1, :], fillrange=X_lo[1, :], fillalpha=0.3,
|
||||||
|
color=:blue, linealpha=0.0, label="reach tube")
|
||||||
|
savefig(fig_n, joinpath(figdir, "reach_operation_n.png"))
|
||||||
|
|
||||||
|
# --- Save result ---
|
||||||
|
matfile = joinpath(@__DIR__, "..", "..", "reachability", "reach_operation_result.mat")
|
||||||
|
matwrite(matfile, Dict("T" => collect(T), "R_lo" => R_lo, "R_hi" => R_hi,
|
||||||
|
"center" => Cnom, "X_lo" => X_lo, "X_hi" => X_hi,
|
||||||
|
"X_nom" => X_nom, "K" => Matrix(K), "A_cl" => A_cl,
|
||||||
|
"delta_entry" => delta_entry,
|
||||||
|
"Q_min" => Q_min, "Q_max" => Q_max,
|
||||||
|
"delta_safe_Tc" => delta_safe_Tc))
|
||||||
|
println("\nSaved reach result to $matfile")
|
||||||
|
|||||||
132
julia-port/src/load_predicates.jl
Normal file
132
julia-port/src/load_predicates.jl
Normal file
@ -0,0 +1,132 @@
|
|||||||
|
"""
|
||||||
|
load_predicates(plant)
|
||||||
|
|
||||||
|
Parse `reachability/predicates.json` into a NamedTuple of concretized
|
||||||
|
halfspaces + constants. Julia counterpart of `reachability/load_predicates.m`.
|
||||||
|
|
||||||
|
Returns a NamedTuple with fields:
|
||||||
|
- `constants` — NamedTuple of plant-derived constants
|
||||||
|
(T_c0, T_f0, T_cold0, T_standby, etc.)
|
||||||
|
- `operational_deadbands` — Dict{Symbol, NamedTuple{(:A_poly, :b_poly, :meaning)}}
|
||||||
|
- `safety_limits` — same shape
|
||||||
|
- `mode_invariants` — NamedTuple with .inv1_holds, .inv2_holds,
|
||||||
|
each holding conjoined A_poly, b_poly,
|
||||||
|
and a components list
|
||||||
|
|
||||||
|
Each `A_poly` is an `n_halfspaces × 10` matrix; the predicate is
|
||||||
|
`{x : A_poly * x <= b_poly}`.
|
||||||
|
|
||||||
|
Multi-coefficient rows are supported: an entry with `"row"` key listing
|
||||||
|
`[[state_index, coeff], ...]` pairs populates multiple columns of one
|
||||||
|
row. Single-coefficient entries (`"state_index"` + `"coeff"`) still
|
||||||
|
work for backward compat.
|
||||||
|
"""
|
||||||
|
function load_predicates(plant)
|
||||||
|
here = @__DIR__
|
||||||
|
json_path = joinpath(here, "..", "..", "reachability", "predicates.json")
|
||||||
|
J = JSON.parsefile(json_path)
|
||||||
|
|
||||||
|
# Context for rhs_expr evaluation.
|
||||||
|
T_c0 = plant.T_c0
|
||||||
|
T_f0 = plant.T_f0
|
||||||
|
T_cold0 = plant.T_cold0
|
||||||
|
T_standby_offset_C = J["derived"]["T_standby_offset_C"]
|
||||||
|
T_standby = T_c0 + T_standby_offset_C
|
||||||
|
|
||||||
|
function eval_rhs(expr::AbstractString)
|
||||||
|
# Tiny sandbox — only the names below are visible.
|
||||||
|
local_ctx = Dict(
|
||||||
|
:T_c0 => T_c0,
|
||||||
|
:T_f0 => T_f0,
|
||||||
|
:T_cold0 => T_cold0,
|
||||||
|
:T_standby => T_standby,
|
||||||
|
)
|
||||||
|
parsed = Meta.parse(expr)
|
||||||
|
# Evaluate within a let-block binding our locals.
|
||||||
|
eval_expr = quote
|
||||||
|
let T_c0 = $T_c0, T_f0 = $T_f0, T_cold0 = $T_cold0, T_standby = $T_standby
|
||||||
|
$parsed
|
||||||
|
end
|
||||||
|
end
|
||||||
|
return eval(eval_expr)
|
||||||
|
end
|
||||||
|
|
||||||
|
function parse_group(group_dict)
|
||||||
|
out = Dict{Symbol, NamedTuple}()
|
||||||
|
for (name, entry) in group_dict
|
||||||
|
startswith(name, "_") && continue
|
||||||
|
!isa(entry, AbstractDict) && continue
|
||||||
|
!haskey(entry, "halfspaces") && continue
|
||||||
|
|
||||||
|
hs_list = entry["halfspaces"]
|
||||||
|
n_hs = length(hs_list)
|
||||||
|
A_poly = zeros(n_hs, 10)
|
||||||
|
b_poly = zeros(n_hs)
|
||||||
|
for (i, hs) in enumerate(hs_list)
|
||||||
|
if haskey(hs, "row")
|
||||||
|
# Multi-coefficient row: [[idx, coeff], ...]
|
||||||
|
for pair in hs["row"]
|
||||||
|
idx = Int(pair[1])
|
||||||
|
coeff = Float64(pair[2])
|
||||||
|
A_poly[i, idx] = coeff
|
||||||
|
end
|
||||||
|
else
|
||||||
|
# Single-coefficient row.
|
||||||
|
A_poly[i, Int(hs["state_index"])] = Float64(hs["coeff"])
|
||||||
|
end
|
||||||
|
b_poly[i] = eval_rhs(hs["rhs_expr"])
|
||||||
|
end
|
||||||
|
meaning = get(entry, "meaning", "")
|
||||||
|
out[Symbol(name)] = (A_poly=A_poly, b_poly=b_poly, meaning=meaning)
|
||||||
|
end
|
||||||
|
return out
|
||||||
|
end
|
||||||
|
|
||||||
|
deadbands = parse_group(J["operational_deadbands"])
|
||||||
|
safety = parse_group(J["safety_limits"])
|
||||||
|
|
||||||
|
# Mode invariants: conjunctions of safety_limits entries.
|
||||||
|
inv_dict = J["mode_invariants"]
|
||||||
|
mode_inv = Dict{Symbol, NamedTuple}()
|
||||||
|
for (name, entry) in inv_dict
|
||||||
|
startswith(name, "_") && continue
|
||||||
|
!isa(entry, AbstractDict) && continue
|
||||||
|
!haskey(entry, "conjunction_of") && continue
|
||||||
|
|
||||||
|
components = entry["conjunction_of"]
|
||||||
|
isa(components, String) && (components = [components])
|
||||||
|
A_all = Matrix{Float64}(undef, 0, 10)
|
||||||
|
b_all = Vector{Float64}()
|
||||||
|
for comp in components
|
||||||
|
sym = Symbol(comp)
|
||||||
|
if haskey(safety, sym)
|
||||||
|
A_all = [A_all; safety[sym].A_poly]
|
||||||
|
b_all = [b_all; safety[sym].b_poly]
|
||||||
|
end
|
||||||
|
end
|
||||||
|
mode_inv[Symbol(name)] = (A_poly=A_all, b_poly=b_all,
|
||||||
|
components=components,
|
||||||
|
meaning=get(entry, "meaning", ""))
|
||||||
|
end
|
||||||
|
|
||||||
|
constants = (
|
||||||
|
T_c0 = T_c0,
|
||||||
|
T_f0 = T_f0,
|
||||||
|
T_cold0 = T_cold0,
|
||||||
|
T_standby = T_standby,
|
||||||
|
T_standby_offset_C = T_standby_offset_C,
|
||||||
|
T_standby_offset_F = J["derived"]["T_standby_offset_F"],
|
||||||
|
t_avg_in_range_halfwidth_C = J["derived"]["t_avg_in_range_halfwidth_C"],
|
||||||
|
p_above_crit_threshold_n = J["derived"]["p_above_crit_threshold_n"],
|
||||||
|
T_fuel_limit_C = J["derived"]["T_fuel_limit_C"],
|
||||||
|
T_c_high_trip_C = J["derived"]["T_c_high_trip_C"],
|
||||||
|
n_high_trip = J["derived"]["n_high_trip"],
|
||||||
|
)
|
||||||
|
|
||||||
|
return (
|
||||||
|
constants = constants,
|
||||||
|
operational_deadbands = deadbands,
|
||||||
|
safety_limits = safety,
|
||||||
|
mode_invariants = mode_inv,
|
||||||
|
)
|
||||||
|
end
|
||||||
77
julia-port/src/pke_solver.jl
Normal file
77
julia-port/src/pke_solver.jl
Normal file
@ -0,0 +1,77 @@
|
|||||||
|
"""
|
||||||
|
pke_solver(plant, Q_sg, ctrl_fn, ref, tspan; x0=nothing, verbose=true)
|
||||||
|
|
||||||
|
Closed-loop simulation of the PKE + thermal-hydraulics plant with a
|
||||||
|
named controller. Analog of `plant-model/pke_solver.m` but using
|
||||||
|
Julia's OrdinaryDiffEq under the hood (Rodas5 for stiff systems,
|
||||||
|
matches MATLAB's ode15s behavior).
|
||||||
|
|
||||||
|
Arguments:
|
||||||
|
- `plant` : parameter NamedTuple from `pke_params()`
|
||||||
|
- `Q_sg` : callable `Q_sg(t)` returning SG heat removal [W]
|
||||||
|
- `ctrl_fn` : callable `u = ctrl_fn(t, x, plant, ref)`
|
||||||
|
- `ref` : any ref struct (NamedTuple or nothing) — passed to controller
|
||||||
|
- `tspan` : `(t_start, t_end)` tuple
|
||||||
|
- `x0` : optional initial state (default: operating-point equilibrium)
|
||||||
|
- `verbose` : print steady-state and final-state banners (default: true)
|
||||||
|
|
||||||
|
Returns `(t, X, U)`:
|
||||||
|
- `t :: Vector{Float64}` — time grid from the solver
|
||||||
|
- `X :: Matrix{Float64}` — M × 10 state matrix (rows are time, cols are states)
|
||||||
|
- `U :: Vector{Float64}` — M-element control trajectory
|
||||||
|
|
||||||
|
The state column indices match `pke_th_rhs!`: `[n, C1..C6, T_f, T_c, T_cold]`.
|
||||||
|
"""
|
||||||
|
function pke_solver(plant, Q_sg, ctrl_fn, ref, tspan; x0=nothing, verbose=true)
|
||||||
|
if x0 === nothing
|
||||||
|
x0 = pke_initial_conditions(plant)
|
||||||
|
end
|
||||||
|
|
||||||
|
CtoF(T) = T * 9 / 5 + 32
|
||||||
|
|
||||||
|
if verbose
|
||||||
|
println("=== Steady-State Conditions ===")
|
||||||
|
@printf " beta = %.5f (%.1f pcm)\n" plant.beta plant.beta*1e5
|
||||||
|
@printf " Lambda = %.1e s\n" plant.Lambda
|
||||||
|
@printf " P0 = %.0f MWth\n" plant.P0/1e6
|
||||||
|
@printf " T_cold = %.1f F\n" CtoF(plant.T_cold0)
|
||||||
|
@printf " T_avg = %.1f F\n" CtoF(plant.T_c0)
|
||||||
|
@printf " T_hot = %.1f F\n" CtoF(plant.T_hot0)
|
||||||
|
@printf " T_fuel = %.1f F\n" CtoF(plant.T_f0)
|
||||||
|
@printf " Core dT = %.1f F\n" plant.dT_core * 9/5
|
||||||
|
println("================================\n")
|
||||||
|
end
|
||||||
|
|
||||||
|
# Closed-loop RHS: controller provides u, then plant RHS.
|
||||||
|
function rhs!(dx, x, p, t)
|
||||||
|
u = ctrl_fn(t, x, plant, ref)
|
||||||
|
pke_th_rhs!(dx, x, t, plant, Q_sg, u)
|
||||||
|
end
|
||||||
|
|
||||||
|
prob = ODEProblem(rhs!, x0, tspan)
|
||||||
|
sol = solve(prob, Rodas5(); reltol=1e-8, abstol=1e-10)
|
||||||
|
|
||||||
|
t_arr = sol.t
|
||||||
|
X = reduce(hcat, sol.u)' # M × 10
|
||||||
|
U = [ctrl_fn(sol.t[k], sol.u[k], plant, ref) for k in 1:length(sol.t)]
|
||||||
|
|
||||||
|
if verbose
|
||||||
|
n_f = X[end, 1]
|
||||||
|
T_f_f = X[end, 8]
|
||||||
|
T_c_f = X[end, 9]
|
||||||
|
T_col_f = X[end, 10]
|
||||||
|
T_hot_f = 2 * T_c_f - T_col_f
|
||||||
|
rho_tot = U[end] + plant.alpha_f*(T_f_f - plant.T_f0) +
|
||||||
|
plant.alpha_c*(T_c_f - plant.T_c0)
|
||||||
|
println("=== Final State (t = $(round(t_arr[end]; digits=1)) s) ===")
|
||||||
|
@printf " Power = %.1f MWth (%.3f x nominal)\n" n_f*plant.P0/1e6 n_f
|
||||||
|
@printf " T_fuel = %.1f F\n" CtoF(T_f_f)
|
||||||
|
@printf " T_hot = %.1f F\n" CtoF(T_hot_f)
|
||||||
|
@printf " T_avg = %.1f F\n" CtoF(T_c_f)
|
||||||
|
@printf " T_cold = %.1f F\n" CtoF(T_col_f)
|
||||||
|
@printf " u = %.4f \$ (external reactivity)\n" U[end]/plant.beta
|
||||||
|
@printf " rho = %.4f \$ (total, incl. T-feedback)\n" rho_tot/plant.beta
|
||||||
|
end
|
||||||
|
|
||||||
|
return t_arr, Matrix(X), U
|
||||||
|
end
|
||||||
50
julia-port/src/plot_pke_results.jl
Normal file
50
julia-port/src/plot_pke_results.jl
Normal file
@ -0,0 +1,50 @@
|
|||||||
|
"""
|
||||||
|
plot_pke_results(t, X, U, plant, Q_sg; title="", savepath=nothing)
|
||||||
|
|
||||||
|
Four-panel visualization of a closed-loop PKE + T/H simulation result.
|
||||||
|
Analog of `plant-model/plot_pke_results.m`. Uses `Plots.jl` with GR
|
||||||
|
backend (default; no extra dependencies).
|
||||||
|
|
||||||
|
Panels:
|
||||||
|
(1) Normalized power and Q_sg demand
|
||||||
|
(2) Coolant temperatures (T_cold, T_avg, T_hot)
|
||||||
|
(3) Fuel temperature
|
||||||
|
(4) External reactivity u (in dollars)
|
||||||
|
"""
|
||||||
|
function plot_pke_results(t, X, U, plant, Q_sg; title="", savepath=nothing)
|
||||||
|
CtoF(T) = T * 9/5 + 32
|
||||||
|
|
||||||
|
n_trace = X[:, 1]
|
||||||
|
T_f = X[:, 8]
|
||||||
|
T_c = X[:, 9]
|
||||||
|
T_cold = X[:, 10]
|
||||||
|
T_hot = 2 .* T_c .- T_cold
|
||||||
|
|
||||||
|
Q_trace = [Q_sg(ti) / plant.P0 for ti in t]
|
||||||
|
u_dollars = U ./ plant.beta
|
||||||
|
|
||||||
|
p1 = plot(t, n_trace, lw=1.8, label="n (normalized power)",
|
||||||
|
xlabel="Time [s]", ylabel="n", legend=:right)
|
||||||
|
plot!(p1, t, Q_trace, lw=1.2, ls=:dash,
|
||||||
|
label="Q_sg / P_0 (demand)")
|
||||||
|
|
||||||
|
p2 = plot(t, CtoF.(T_cold), lw=1.8, label="T_cold",
|
||||||
|
xlabel="Time [s]", ylabel="T [F]", legend=:right)
|
||||||
|
plot!(p2, t, CtoF.(T_c), lw=1.8, label="T_avg")
|
||||||
|
plot!(p2, t, CtoF.(T_hot), lw=1.8, label="T_hot")
|
||||||
|
|
||||||
|
p3 = plot(t, CtoF.(T_f), lw=1.8, label="T_fuel",
|
||||||
|
xlabel="Time [s]", ylabel="T [F]", legend=:right)
|
||||||
|
|
||||||
|
p4 = plot(t, u_dollars, lw=1.8, label="u [\$]",
|
||||||
|
xlabel="Time [s]", ylabel="Reactivity [\$]", legend=:right)
|
||||||
|
|
||||||
|
fig = plot(p1, p2, p3, p4, layout=(2,2), size=(1100, 700),
|
||||||
|
plot_title=title, plot_titlefontsize=11)
|
||||||
|
|
||||||
|
if savepath !== nothing
|
||||||
|
savefig(fig, savepath)
|
||||||
|
end
|
||||||
|
|
||||||
|
return fig
|
||||||
|
end
|
||||||
71
julia-port/src/reach_linear.jl
Normal file
71
julia-port/src/reach_linear.jl
Normal file
@ -0,0 +1,71 @@
|
|||||||
|
"""
|
||||||
|
reach_linear(A_cl, B_w, x0_center, x0_halfwidth, w_lo, w_hi, tspan, dt)
|
||||||
|
|
||||||
|
Hand-rolled box-zonotope reach tube for `dx/dt = A_cl*x + B_w*w` with
|
||||||
|
`x(0)` in an axis-aligned box (center, halfwidths) and `w` in a scalar
|
||||||
|
interval. Julia port of `reachability/reach_linear.m`.
|
||||||
|
|
||||||
|
Returns `(T, R_lo, R_hi, center)`:
|
||||||
|
- `T :: Vector{Float64}` — time grid
|
||||||
|
- `R_lo :: Matrix{Float64}` — `n × M` lower bounds
|
||||||
|
- `R_hi :: Matrix{Float64}` — `n × M` upper bounds
|
||||||
|
- `center :: Matrix{Float64}` — `n × M` nominal (w_mid-driven) trajectory
|
||||||
|
|
||||||
|
Mathematical setup:
|
||||||
|
x(t) = e^(A·t)·x(0) + integral_0^t e^(A·(t-s))·B_w·w(s) ds
|
||||||
|
Per step of size dt:
|
||||||
|
A_step = expm(A_cl * dt)
|
||||||
|
G_step = integral_0^dt e^(A_cl·s)·B_w ds (via Van Loan augmented trick)
|
||||||
|
Halfwidths propagate using elementwise-|·| matrix: a box under linear
|
||||||
|
map M becomes a box with halfwidth |M|·h (sound over-approximation).
|
||||||
|
"""
|
||||||
|
function reach_linear(A_cl, B_w, x0_center, x0_halfwidth, w_lo, w_hi, tspan, dt)
|
||||||
|
n = size(A_cl, 1)
|
||||||
|
@assert size(A_cl, 2) == n "A_cl must be square"
|
||||||
|
x0_center = vec(x0_center)
|
||||||
|
x0_halfwidth = vec(x0_halfwidth)
|
||||||
|
B_w_col = vec(B_w)
|
||||||
|
|
||||||
|
w_mid = 0.5 * (w_lo + w_hi)
|
||||||
|
w_half = 0.5 * (w_hi - w_lo)
|
||||||
|
|
||||||
|
T = collect(tspan[1]:dt:tspan[2])
|
||||||
|
M = length(T)
|
||||||
|
|
||||||
|
R_lo = zeros(n, M)
|
||||||
|
R_hi = zeros(n, M)
|
||||||
|
centr = zeros(n, M)
|
||||||
|
|
||||||
|
A_step = exp(A_cl * dt)
|
||||||
|
A_abs = abs.(A_step)
|
||||||
|
|
||||||
|
# Van Loan augmented matrix: upper-right block of exp([A B_w; 0 0]*dt)
|
||||||
|
# equals integral_0^dt exp(A*s)*B_w ds.
|
||||||
|
aug = [A_cl B_w_col; zeros(1, n+1)]
|
||||||
|
aug_exp = exp(aug * dt)
|
||||||
|
G_step = aug_exp[1:n, n+1]
|
||||||
|
G_abs = abs.(G_step)
|
||||||
|
|
||||||
|
x_center_now = copy(x0_center)
|
||||||
|
halfwidth_now = copy(x0_halfwidth)
|
||||||
|
S_center = zeros(n)
|
||||||
|
S_half = zeros(n)
|
||||||
|
|
||||||
|
centr[:, 1] = x_center_now .+ S_center
|
||||||
|
R_lo[:, 1] = centr[:, 1] .- halfwidth_now .- S_half
|
||||||
|
R_hi[:, 1] = centr[:, 1] .+ halfwidth_now .+ S_half
|
||||||
|
|
||||||
|
for k in 2:M
|
||||||
|
x_center_now = A_step * x_center_now
|
||||||
|
halfwidth_now = A_abs * halfwidth_now
|
||||||
|
|
||||||
|
S_center = A_step * S_center + G_step * w_mid
|
||||||
|
S_half = A_abs * S_half + G_abs * w_half
|
||||||
|
|
||||||
|
centr[:, k] = x_center_now .+ S_center
|
||||||
|
R_lo[:, k] = centr[:, k] .- halfwidth_now .- S_half
|
||||||
|
R_hi[:, k] = centr[:, k] .+ halfwidth_now .+ S_half
|
||||||
|
end
|
||||||
|
|
||||||
|
return T, R_lo, R_hi, centr
|
||||||
|
end
|
||||||
Loading…
x
Reference in New Issue
Block a user