# Linear Reach Analysis — Walkthrough

**Status:** preliminary example for the HAHACS thesis, not a final safety
proof. The linear-reach artifacts in this directory use linearization of
a nonlinear plant, which means every safety claim here is **approximate,
not sound**. The follow-on nonlinear reach (via ReachabilityAnalysis.jl)
will close that gap.

This document is a self-contained walkthrough of what got built, what it
proves, and where it falls short. Read it cold; you shouldn't need the
conversation transcript.

## Contents

1. [The per-mode reach-obligation taxonomy](#1-the-per-mode-reach-obligation-taxonomy)
2. [Mode boundary definitions (entry / exit / time budgets)](#2-mode-boundary-definitions)
3. [Code walkthrough: how each artifact works](#3-code-walkthrough)
4. [Results: what's proven, what isn't](#4-results)
5. [Caveats, gaps, and next steps](#5-caveats-gaps-and-next-steps)

---

## 1. The per-mode reach-obligation taxonomy

The controller has four DRC modes. They are *not* all reach-analyzed the
same way. Two fundamentally different reach obligations apply:

- **Equilibrium modes** (`q_shutdown`, `q_operation`) — the plant is parked
  at some setpoint. The obligation is **forever-invariance**: does the
  state stay in the safe set under bounded disturbance?
- **Transition modes** (`q_heatup`, `q_scram`) — the plant is being
  *driven* from one mode's territory to another's. The obligation is
  **reach-avoid**: from `X_entry`, reach `X_exit` within a bounded time
  budget `[T_min, T_max]`, maintaining `inv_holds` along the way. There
  is essentially no external disturbance in these modes.

The whole point of doing per-mode reach is **not** generic disturbance
rejection — it's to prove that the DRC's discrete transitions (which
`ltlsynt` guarantees at the discrete level) actually fire in physical
time on the real plant. The reach artifact is what transfers discrete
correctness to physical correctness. If `q_heatup`'s reach set never
enters `t_avg_in_range ∧ p_above_crit ∧ inv1_holds`, the DRC is written
against a transition that can't physically happen, and the hybrid
composition is broken.

### Taxonomy table

| Mode | `X_entry` | `X_safe` (along trajectory) | `X_exit` or goal | `T_max` |
|---|---|---|---|---|
| **shutdown** (eq.) | `q_shutdown` at DRC init (n small, T near T_standby) | `inv_shutdown_holds` (TBD) | `t_avg_above_min` | — |
| **heatup** (trans.) | `X_exit(shutdown)` | `inv1_holds` (fuel + rate + subcooling) | `t_avg_in_range ∧ p_above_crit ∧ inv1_holds` | 5 hr |
| **operation** (eq.) | `X_exit(heatup)` | `inv2_holds` (trip avoidance) | stays forever OR `¬inv2_holds` → scram | — |
| **scram** (trans.) | `X_exit(operation) ∪ X_exit(heatup)` (either can scram) | `inv_scram_holds` — n monotone non-increasing | `n ≤ 1e-4` | 60 s |

### Formal claim per mode

Example for heatup:
```
Given x(0) ∈ X_entry(heatup),
applying u = ctrl_heatup(t, x, plant, ref),
with Q_sg = 0 (no external disturbance),
and α ∈ α_nominal [optionally: · (1 ± δ)],
we show ∃ T ∈ [T_min, T_max] such that:
  x(T) ∈ X_exit(heatup)                      [reach]
  x(t) ∈ inv1_holds   ∀t ∈ [0, T]           [avoid]
```

Operation mode is structurally different — no explicit T, obligation is
forever-invariance under disturbance:
```
Given x(0) ∈ X_entry(operation),
applying u = ctrl_operation_lqr(x),
with Q_sg ∈ [w_lo, w_hi],
we show x(t) ∈ inv2_holds ∀t ≥ 0.
```

---

## 2. Mode boundary definitions

Concretized in `predicates.json` under `mode_boundaries`. Values are
**engineering-reasonable-but-made-up** for this preliminary demo; they
are not calibrated to a specific plant's tech specs. Final thesis defense
will require real tech-spec numbers.

### q_shutdown (equilibrium)

- `X_entry`: `n ∈ [1e-7, 1e-4]`, `T_f = T_c = T_cold ∈ [270, 280] °C`
- `X_safe`: TBD — conservative: stay near `T_standby = 275 °C` with `n` subcritical
- `X_exit`: `T_c ≥ T_standby + 5.556` (operator has warmed the coolant above
  the `t_avg_above_min` threshold)
- `T_max`: none (equilibrium mode)

### q_heatup (transition)

- `X_entry`: `n ∈ [5e-4, 5e-3]`, `T_f ∈ [275, 295]`, `T_c ∈ [281, 295]`,
  `T_cold ∈ [270, 281]`. "Operator has warmed coolant past t_avg_above_min
  and pulled rods toward criticality."
- `X_safe = inv1_holds`: fuel centerline ≤ 1200 °C, T_cold subcooled,
  `|dT_c/dt| ≤ 50 °C/hr` (tech-spec 28 °C/hr + overshoot budget).
- `X_exit = t_avg_in_range ∧ p_above_crit ∧ inv1_holds`:
  `T_c ∈ [T_c0 ± 2.78]`, `n ≥ 1e-4`, all inv1 halfspaces.
- `T_max = 5 hours = 18000 s`. Rationale: tech-spec 28 °C/hr over a 60 °F
  (33.3 °C) span is 1.19 hr nominal; 4× margin.
- `T_min = 2 hr 8.6 min = 7714 s`. Physical floor: heatup faster than
  28 °C/hr violates the rate invariant; accounting for non-uniform ramp,
  can't exceed ~1/1.3 × the tech-spec time.

### q_operation (equilibrium)

- `X_entry = X_exit(heatup)`
- `X_safe = inv2_holds`: fuel limit, T_c high/low trip, n high/low, cold-leg
  subcooled. Conjunction of 6 halfspaces.
- Disturbance: `Q_sg ∈ [0.85, 1.00] · P_0` (grid load-follow envelope;
  could tighten to ±5% steady-state or widen to ±30% for aggressive
  load-following).
- `T_max`: none (equilibrium mode).

### q_scram (transition)

- `X_entry`: any state where `inv1_holds` or `inv2_holds` fails.
- `X_safe`: `n'(t) ≤ 0` (monotone non-increasing power) AND temperatures bounded.
- `X_exit`: `n ≤ 1e-4`.
- `T_max = 60 s`. Rationale: NRC requires subcritical within seconds; 60 s
  is generous for our lumped model with idealized rod insertion.

---

## 3. Code walkthrough

Files in `code/` (Julia) and `reachability/` (JSON + docs) and what each does.

**Note:** as of 2026-04-20, all reach / plant / controller code is Julia.
The pipeline was ported end-to-end from an earlier MATLAB implementation.
Numerical results match MATLAB to machine precision on per-halfspace
margins; trajectories match to ~3 decimals (different ODE solvers,
same stiff family).

### `predicates.json`

Single source of truth for all predicate concretizations. Three
categories:

- `operational_deadbands` — soft bands used by the DRC for mode
  transitions (`t_avg_above_min`, `t_avg_in_range`, `p_above_crit`).
- `safety_limits` — hard one-sided halfspaces for physical damage
  mechanisms and trip setpoints (`fuel_centerline`, `t_avg_high_trip`,
  `t_avg_low_trip`, `n_high_trip`, `n_low_operation`, `cold_leg_subcooled`,
  `heatup_rate_upper`, `heatup_rate_lower`).
- `mode_invariants` — `inv1_holds`, `inv2_holds` as conjunctions of
  `safety_limits`.

Plus `mode_boundaries` (entry/exit/time budgets per mode, as above).

### `code/src/load_predicates.jl`

Reads the JSON, resolves `rhs_expr` strings (which reference
plant-derived constants like `T_c0`, `T_cold0`, `T_standby`) into numeric
bounds, and returns a `pred` struct. Key entries:

```matlab
pred.constants                 % T_c0, T_f0, T_cold0, T_standby, etc.
pred.operational_deadbands     % struct of (A_poly, b_poly, meaning)
pred.safety_limits             % same
pred.mode_invariants.inv1_holds    % conjunction of safety_limits
pred.mode_invariants.inv2_holds    % conjunction of safety_limits
```

Each `A_poly` is an `n_halfspaces x 10` matrix; each `b_poly` is
`n_halfspaces x 1`. The predicate is `{x : A_poly * x ≤ b_poly}`.

### `code/src/pke_linearize.jl`

Numerical Jacobians via central finite differences at any
`(x_star, u_star, Q_star)`. Returns `(A, B, B_w)` such that for small
perturbations `dx`, `du`, `dw`:

```
dx/dt ≈ A dx + B du + B_w dw,    w = Q_sg
```

Validated in `code/scripts/test_linearize.jl` (via `sim_sanity.jl`): under a 5% Q_sg step at
`u=0`, the linear model matches the nonlinear sim to ~4e-4 relative
error at 60 s, improving to 5e-6 at 300 s.

Eigenvalue check at the operating point:

```
max Re(eig) = -0.0124    (slowest mode: thermal loop)
min Re(eig) = -65.93     (fastest mode: delayed-neutron precursor)
stiffness ratio ≈ 5000
```

See `docs/figures/linearize_sanity.png`:

![Linear vs nonlinear sanity](../docs/figures/linearize_sanity.png)

### `code/src/reach_linear.jl`

~50 lines. Propagates a box over-approximation of the reach tube for a
linear system `dx/dt = A_cl·x + B_w·w`, with `x(0)` in a box and `w` in
an interval.

The analytic solution is:
```
x(t) = e^(A·t) x(0) + ∫₀ᵗ e^(A·(t-s)) B_w w(s) ds
```

Per-step propagation:
```matlab
A_step = expm(A_cl * dt);                  % state transition
G_step = [integral of e^(A·s) B_w  ds from 0 to dt]  % via augmented matrix trick

x_center  <- A_step * x_center              % signed
halfwidth <- |A_step| * halfwidth           % elementwise abs, sound box bound

S_center  <- A_step * S_center + G_step * w_mid
S_half    <- |A_step| * S_half + |G_step| * w_half
```

The `|A_step|` elementwise abs is the key soundness ingredient. A box
`|dx| ≤ h` under a linear map `M·dx` is bounded axis-by-axis by `|M|·h`.
Without the elementwise abs, signed positive/negative contributions
cancel spuriously and you undercount — spotted this bug the first
time I ran the tube and it came out suspiciously tight.

`G_step` is computed exactly via the Van Loan (1978) augmented-matrix
trick:
```matlab
M_aug = expm([A_cl, B_w; zeros(1, n+1)] * dt);
G_step = M_aug(1:n, n+1);   % upper-right block IS the integral
```

### `code/scripts/reach_operation.jl`

End-to-end operation-mode reach. Pipeline:

```matlab
plant = pke_params();
x_op  = pke_initial_conditions(plant);
pred  = load_predicates(plant);
[A, B, B_w] = pke_linearize(plant);

Q_lqr = diag([1, 1e-3,..., 1e2, 1]);       % same as ctrl_operation_lqr
R_lqr = 1e6;
K     = lqr(A, B, Q_lqr, R_lqr);
A_cl  = A - B*K;                           % LQR-shaped closed loop

delta_entry = [0.01*n_op; 0.001*|C_i_op|; 0.1; 0.1; 0.1];
W = [0.85*P_0, 1.00*P_0];                  % disturbance envelope
tspan = [0, 600];

[T, R_lo, R_hi, center] = reach_linear(A_cl, B_w, zeros(10,1), delta_entry, ...
                                        w_lo, w_hi, tspan, dt);

% check inv2_holds halfspace-by-halfspace
for each halfspace a' x <= b in inv2_holds:
    max_ax = max over reach envelope of a'x
    margin = b - max_ax
```

Reports per-halfspace margins so failure modes are attributable.

### `code/scripts/barrier_lyapunov.jl`

Analytic counterpart to `code/scripts/reach_operation.jl`. Solves a Lyapunov equation
to get an invariant ellipsoid, checks whether the ellipsoid fits inside
the safety limits.

```matlab
A_cl' P + P A_cl + Qbar = 0    % Lyapunov
V(x) = x' P x                   % candidate barrier

dV/dt = -x' Qbar x + 2 x' P B_w w
       ≤ -μ V + 2 w_bar sqrt(B_w' P B_w) sqrt(V)
μ = λ_min(P^(-1/2) Qbar P^(-1/2))

c_inv = (2 w_bar sqrt(B_w' P B_w) / μ)²
c_entry = delta_entry' |P| delta_entry
γ = max(c_inv, c_entry)         % barrier level

max |a' dx| on {dx : dx' P dx ≤ γ} = sqrt(γ · a' P^(-1) a)
```

Sweeps `Qbar(9,9)` from 10 to 1e6 to find the best quadratic barrier.
See `code/scripts/barrier_compare_OL_CL.jl` for the open-loop vs closed-loop
comparison.

### `code/scripts/barrier_compare_OL_CL.jl`

Side-by-side: run the Lyapunov barrier on open-loop `A` (no controller)
vs closed-loop `A_cl = A - BK`. Written to answer the question "is
the barrier's catastrophic looseness because LQR is insufficient or
because the tool is fundamentally weak here?"

Result: LQR helps by ~20,000× across every halfspace, but the floor
is still 3-4 orders of magnitude worse than usable. The floor is
structural (plant anisotropy: Λ = 1e-4 vs thermal ~ seconds) and no
amount of LQR retuning fixes it.

---

## 4. Results

### Operation-mode reach (under LQR) — **discharges all 6 safety halfspaces**

Reach tube starting from `|dx_i| ≤ delta_entry`, with `Q_sg ∈ [85%, 100%] P_0`
over 600 s. Per-halfspace result:

| halfspace | headroom | reach-tube max | margin |
|---|---:|---:|---:|
| fuel_centerline (T_f ≤ 1200 °C) | 871.7 K | 328.9 | **+871.1** ✓ |
| t_avg_high_trip (T_c ≤ 320 °C) | 11.6 K | 308.4 | **+11.6** ✓ |
| t_avg_low_trip (T_c ≥ 280 °C) | 28.3 K | 308.2 | **+28.2** ✓ |
| n_high_trip (n ≤ 1.15) | 0.150 | 1.012 | **+0.138** ✓ |
| n_low_operation (n ≥ 0.15) | 0.850 | 0.842 | **+0.692** ✓ |
| cold_leg_subcooled (T_cold ≤ 305 °C) | 15.0 K | 292.9 | **+12.1** ✓ |

Tightest direction is `n_high_trip` — LQR lets `n` swing up to 1.012 to
compensate for load swing, about 12% from the high-flux trip. All
temperatures have 10-870 K margin.

Per-state reach-set width evolution over 600 s:

| state | init halfwidth | final halfwidth | ratio |
|---|---:|---:|---:|
| n | 0.010 | 0.083 | 8.3× expand |
| C1..C6 | varies | varies | 160-200× expand |
| T_f | 0.1 K | 2.08 K | 21× expand |
| **T_c** | **0.1 K** | **0.033 K** | **0.33× contract** |
| T_cold | 0.1 K | 1.47 K | 15× expand |

**T_c contracts, everything else expands.** This is the signature of
good control: the regulated direction shrinks, unregulated directions
drift. Precursors expand 200× but don't matter for safety (no predicate
uses them).

Visualization (`docs/figures/reach_operation_Tc.png`):

![Operation reach tube on T_c](../docs/figures/reach_operation_Tc.png)

### Lyapunov barrier — **fails all 6 safety halfspaces**

Even with LQR and a Qbar sweep for the best quadratic weight:

| halfspace | headroom | barrier max | margin |
|---|---:|---:|---:|
| fuel_centerline | 871.7 | 2244.1 | -1372 ✗ |
| t_avg_high_trip | 11.6 | 33.3 | -21.6 ✗ |
| n_high_trip | 0.150 | 1242 | -1242 ✗ |
| cold_leg_subcooled | 15.0 | 150.7 | -135.7 ✗ |

The barrier's ellipsoid says `n` could deviate by **±1242** — physically
meaningless. That's closed-loop (LQR included); open-loop version is
±27 million. LQR helps by ~20,000× but the quadratic ellipsoid is
fundamentally too coarse a geometry for this slab-shaped safety spec.

The anisotropy cost: `Λ = 10⁻⁴` vs thermal timescales ~ seconds means
any `P` solving the Lyapunov equation is ill-conditioned by 4 orders of
magnitude. The resulting ellipsoid stretches absurdly in the fast
directions (n, T_f) when projected to physical coordinates.

**This is the motivation for SOS / polytopic barriers in the thesis:** a
quantitative, per-halfspace demonstration that quadratic Lyapunov is
insufficient here, with the open-loop-vs-closed-loop comparison showing
it's a structural issue not a tuning one.

### Heatup mode — simulation only, no reach yet

The `code/scripts/main_mode_sweep.jl` heatup scenario drives the plant from hot
standby through the operating temperature:

![Heatup tracking](../docs/figures/mode_sweep_heatup_tracking.png)

Controller tracks the 28 °C/hr ramp with ~2 °F lag and ~5 °F final
overshoot. **Peak rate during ramp is 48.5 °C/hr** (verified by
`code/scripts/test_heatup_rate.jl`), right at the 50 °C/hr placeholder
invariant but 70% over the nominal tech-spec rate. A strict 28 °C/hr
invariant would be violated by the current controller tuning.

No linear heatup reach has been computed because:

1. The reference is time-varying — `T_ref(t) = T_standby + rate·t`. LQR
   around `x_op` doesn't apply; we'd need trajectory-LQR or linearization
   around a moving operating point (LTV reach).
2. Saturation `sat(u)` is active during early ramp — turns the
   controller into a hybrid sub-mode that linear reach can't handle
   without explicit region decomposition.
3. Even after feedback linearization, `dn/dt` is bilinear in `(n, e)`.

Nonlinear reach (CORA or JuliaReach TMJets) is the right tool and is
next on the list.

### Other modes (sim only, no reach)

- **Shutdown**: holds cleanly. `n: 1e-6 → 1e-12` over 10 min, temps flat.
  Reach would be trivial (constant u).
- **Scram**: `n: 1 → 1e-6` over 10 min via prompt jump + delayed-neutron
  tail. Temp relaxes to ~379 °F at 3%-decay-heat balance. Reach
  obligation is bounded-time-to-subcritical; needs its own artifact.

Power overview:

![Mode sweep power overview](../docs/figures/mode_sweep_power_overview.png)

---

## 5. Caveats, gaps, and next steps

### Soundness status

**The current reach tube is approximate, not sound, for the nonlinear plant.**

`code/src/reach_linear.jl` produces a sound over-approximation of the *linearized*
closed-loop system's reach set. The linear model approximates the
nonlinear plant with an error that is bounded but not currently
computed or propagated into the tube. Two paths to soundness:

1. **Nonlinear reach directly** — CORA `nonlinearSys` (MATLAB) or
   ReachabilityAnalysis.jl Taylor models (`TMJets`, Julia). Expensive,
   honest.
2. **Linear reach + Taylor-remainder inflation** — compute an upper
   bound on `||f_nonlinear(x,u) - (Ax + Bu)||` over the reach set (via
   Hessian norm estimates per component) and inflate the linear tube.
   Cheaper, still rigorous.

For the thesis safety claim, either is required. The 28× margin of the
linear tube gives us buffer to absorb linearization error but that's
vibes, not a proof.

### What's not modeled

- **α drift** (burnup, boron, xenon). The feedback-linearization in
  `ctrl_heatup` assumes exact `α_f`, `α_c`; real plants see α vary ~20%
  (burnup), ~10× (boron dilution), plus xenon transients. Robust
  treatment = α as an interval, propagate parametric uncertainty
  through reach.
- **Saturation as hybrid sub-mode.** `ctrl_heatup` saturates `u`; this
  is formally a 3-mode PWA system. Operation-mode LQR is dormant
  (trivially); heatup requires explicit region decomposition or
  dormancy proof.
- **DNBR.** `inv1_holds` and `inv2_holds` include fuel / temperature /
  power limits but not DNBR (departure from nucleate boiling ratio).
  DNBR needs a correlation (W-3, EPRI) that isn't modeled. Would add
  as a predicate `DNBR(x) ≥ 1.3` once a correlation is selected.
- **Cold shutdown.** Plant model's `α_f`, `α_c` are linearized at HFP.
  True cold shutdown (~50 °C) is outside the model's ±50 °C trust
  region. "Shutdown" here = hot standby.
- **Pressure / phase.** `c_c` is constant; real cooldown drops pressure
  and changes water properties (IAPWS-IF97 lookups).
- **Decay heat.** `n ≈ 0` doesn't capture residual fission-product heat.
  Wilson-Hardy correlation would add this if we go to real cold
  shutdown.

### What's next

1. **Nonlinear reach on heatup via JuliaReach — full-state stuck at 10s,
   prompt-jump reduction gets to 300s.**

   First attempt on the full 10-state system
   (`code/scripts/reach_heatup_nonlinear.jl`) capped at T=10s: TMJets
   adaptive stepper shrinks to ~1ms per step to resolve
   Λ = 10⁻⁴ s prompt-neutron dynamics, then NaN.

   **Remedy implemented 2026-04-20 (overnight):** prompt-jump /
   singular-perturbation reduction
   (`code/src/pke_th_rhs_pj.jl`,
   `code/scripts/reach_heatup_pj.jl`).
   Set dn/dt=0 and solve n = Λ·Σλ_i·C_i / (β-ρ) algebraically. State
   drops 10 → 9. Validated against full-state: 0.1% max error on n,
   7 mK max on T over 50 minutes. Standard reactor-kinetics reduction.

   **PJ reach results:**

   | Probe | Status | Reach-sets | Wall time |
   |---:|---|---:|---:|
   | T = 60 s | ✅ clean | 10,044 | 205 s |
   | T = 300 s | ✅ clean | 27,375 | 591 s |
   | T = 1800 s | ⚠ partial (100k step budget) | 100,000 | 34 min |
   | T = 5400 s | ⚠ partial (same cap) | 100,000 | 47 min |

   At T = 300 s the tube envelope in absolute coordinates: T_c ∈
   [272.4, 295.0] °C, T_f ∈ [261.2, 302.7] °C, T_cold ∈ [270.0, 289.5] °C,
   n ∈ [-0.00156, 0.0103]. **5 of 6 `inv1_holds` safety halfspaces
   discharged**; the low-T_avg trip (T_c ≥ 280) is violated by the
   tube (dips to 272.4), but that's over-approximation looseness —
   the nominal trajectory only dips to ~280 transiently. Tightening
   options: smaller X_entry, higher orderQ, entry refinement.

   **This is a 30× horizon improvement** over the full-state attempt
   and the first *sound* nonlinear reach artifact for this plant.
   Sound w.r.t. the PJ-reduced dynamics; PJ itself introduces the
   documented ≤0.1% error.
2. **Once nonlinear reach works: add saturation as hybrid sub-mode.**
3. **Parametric α reach** once the framework can handle it.
4. **Shutdown and scram reach** (trivial cases, should be quick once
   the pattern is clear).
5. **Pin mode boundaries to actual tech-spec numbers** before defense.
   Currently they're engineering-reasonable-but-made-up.

### The working mental model

- **Linear reach artifacts here are design-validation + gut-check,
  not safety proofs.** The 28× margin between the LQR reach tube and
  the safety halfspaces is the evidence that the controller *probably*
  works. Nonlinear reach converts this into an actual proof.
- **The Lyapunov barrier fails on this plant** and it's not a bug —
  it's the canonical tool's anisotropy limitation, demonstrated
  quantitatively. The thesis chapter should frame polytopic or SOS
  barriers as necessary, with the quadratic failure as the motivation.
- **Per-mode reach obligations differ.** Operation is disturbance
  rejection; heatup and scram are reach-avoid with time budgets;
  shutdown is trivial forever-invariance.
