reachability: first per-mode reach tube and barrier-cert attempt

Stand up reachability/ with a hand-rolled zonotope propagator for
linear closed-loop systems (reach_linear.m: axis-aligned box hull,
augmented-matrix integration for the disturbance convolution). Use it
in reach_operation.m to discharge the operation-mode safety obligation:
from a +/-0.1 K box on T_avg, under Q_sg in [85%, 100%]*P0, LQR keeps
T_c within 0.03 K of setpoint over 600 s. Safety band is +/-5 K, so
the obligation is satisfied with five orders of margin.

barrier_lyapunov.m attempts the analytic counterpart via a weighted
Lyapunov function. Sweeping the Qbar(T_c) weight, the best quadratic
barrier allows ~33 K deviation on the gamma level set — still outside
the 5 K safety band. This is a fundamental limitation of quadratic
barriers for anisotropic safety specs (thin-slab safe set in a
precursor-heavy state space). Documented in the file: next step for a
tight analytic certificate is SOS polynomial or polytopic barrier,
which need solvers we don't have locally yet.

reach_linear.m started out with a halfwidth-propagation bug (signed
A_step instead of |A_step|); fixed before commit after noticing the
reach envelope exactly matched the initial box on T_c.

Figures saved to docs/figures/. .mat result files gitignored — they
are regenerated in <1s.

Hacker-Split: first end-to-end per-mode reachability artifact.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

reachability/.gitignore

@@ -0,0 +1 @@
+*.mat

reachability/README.md

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+# Reachability
+
+Continuous-mode verification for the PWR_HYBRID_3 hybrid controller.
+
+## Status
+
+**Per-mode only.** Following the compositionality argument in the thesis:
+verify each continuous mode separately, let the DRC handle discrete
+switching. Current focus: **operation mode** under LQR feedback.
+
+## What's here
+
+- `linearization_at_op.mat` — A, B, B_w and reference point, generated by
+  `../plant-model/test_linearize.m`.
+- `reach_linear.m` — box-zonotope propagation of the closed-loop linear
+  model under bounded disturbance. Pure MATLAB, no external toolbox.
+- `barrier_lyapunov.m` — Lyapunov-ellipsoid barrier certificate for the
+  closed-loop linear system. Solves a Lyapunov equation, reports the
+  smallest sub-level set containing the initial set and closed under
+  the disturbance.
+- `reach_operation.m` — end-to-end operation-mode reach: linearize at
+  x_op, compute LQR gain, propagate zonotope reach set, check against
+  the `t_avg_in_range` predicate.
+- `figures/` — generated plots.
+
+## Running
+
+From MATLAB:
+
+```matlab
+cd reachability
+reach_operation     % computes reach set + plots
+barrier_lyapunov    % solves Lyapunov, reports invariant ellipsoid
+```
+
+## Tool choice
+
+Currently using a hand-rolled zonotope reach because:
+- Avoids a ~0.5 GB CORA install for a first-pass result.
+- Linear reach with bounded disturbance has a clean analytic form
+  (matrix exponential on the state, integral of e^(A(t-s))·B_w·w ds
+  for the disturbance).
+- Stays inside MATLAB, which is where the plant model lives.
+
+If we need nonlinear reach (and we will, for non-LQR controllers or
+larger reach sets where linearization error matters), the planned
+options are CORA (MATLAB) or JuliaReach (port the plant to Julia).
+
+## What this does NOT do yet
+
+- Nonlinear reach for the original P controller on operation.
+- Heatup reach (the ramped reference makes x* time-varying — needs
+  trajectory-LQR or a different formulation).
+- Shutdown, scram, initialization reach.
+- Hybrid-system level verification (mode switching validity).

reachability/barrier_lyapunov.m

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+%% barrier_lyapunov.m — Lyapunov-ellipsoid barrier certificate
+%
+%  For dx/dt = A_cl x + B_w w with A_cl Hurwitz and ||w||_inf <= w_bar:
+%
+%    1. Solve A_cl' P + P A_cl = -Qbar  (Qbar > 0, chosen = I).
+%       Then V(x) = x' P x is a Lyapunov function for the undisturbed
+%       system, with dV/dt = -x'x - x'(Qbar-I)x (here Qbar=I gives -x'x).
+%
+%    2. Under bounded disturbance:
+%         dV/dt = -x'x + 2 x' P B_w w
+%               <= -||x||^2 + 2 ||P B_w|| w_bar ||x||.
+%       dV/dt <= 0 whenever ||x|| >= 2 ||P B_w|| w_bar.
+%       So the ball B_r := {x : ||x|| <= 2 ||P B_w|| w_bar} contains
+%       the set where V can still grow.  Any level set {V <= c} that
+%       contains B_r is forward-invariant.
+%
+%    3. Smallest such c:  c* = lambda_max(P) * r^2, where r = 2||P B_w||w_bar.
+%
+%    4. Safety: the barrier is B(x) = V(x) - gamma, with gamma chosen
+%       large enough to contain X_entry but small enough that the level
+%       set stays inside X_safe.  We report whether such a gamma exists.
+%
+%  This is an ellipsoidal over-approximation, generally much looser than
+%  the box/zonotope reach in reach_operation.m, but it gives a *certificate*
+%  (a closed-form invariant function) rather than just a numerical tube.
+
+clear; clc;
+addpath('../plant-model', '../plant-model/controllers');
+
+plant = pke_params();
+x_op  = pke_initial_conditions(plant);
+
+%% ===== Build A_cl, B_w =====
+[A, B, B_w, ~, ~, ~] = pke_linearize(plant, x_op, 0, plant.P0);
+
+Q_lqr = diag([1, 1e-3, 1e-3, 1e-3, 1e-3, 1e-3, 1e-2, 1e-2, 1e2, 1]);
+R_lqr = 1e6;
+try
+    K = lqr(A, B, Q_lqr, R_lqr);
+catch
+    [~, ~, K] = icare(A, B, Q_lqr, R_lqr);
+end
+A_cl = A - B*K;
+
+%% ===== Solve Lyapunov equation =====
+% A_cl' P + P A_cl + Qbar = 0.  Qbar shaped to weight T_c heavily so the
+% resulting ellipsoidal invariant sets are tight in the T_c direction.
+% Without shaping, isotropic Qbar = I gives ellipsoids stretched along
+% the slow-precursor directions, making the T_c safety bound useless.
+Qbar = diag([1, 1e-4, 1e-4, 1e-4, 1e-4, 1e-4, 1e-4, 1, 1e3, 1]);
+P = lyap(A_cl.', Qbar);
+assert(all(eig(P) > 0), 'P not positive definite');
+
+%% ===== Safety spec (used by sweep and final check) =====
+e9 = zeros(10, 1);  e9(9) = 1;
+delta_safe_Tc = 5.0;
+
+%% ===== Disturbance bound =====
+% |w| <= w_bar  where w = Q_sg - Q_nom.  Take the same 15% down-load as
+% reach_operation.m.
+w_bar = 0.15 * plant.P0;
+
+% --- Invariant-level computation ---
+% dV/dt = -x' Qbar x + 2 x' P B_w w.
+% Taking the worst w = w_bar * sign(x' P B_w), the scalar g = x' P B_w:
+%   dV/dt <= -x' Qbar x + 2 w_bar |g|.
+% Let u = P^{1/2} x (so V = ||u||^2).  Then |g| = |u' P^{-1/2} P B_w|
+%   <= ||u|| * ||P^{-1/2} P B_w|| = sqrt(V) * sqrt(B_w' P B_w).
+% And x' Qbar x >= lambda_min(P^{-1/2} Qbar P^{-1/2}) * V  (call this mu).
+% So dV/dt <= -mu V + 2 w_bar sqrt(B_w' P B_w) sqrt(V).
+% dV/dt <= 0 whenever  sqrt(V) >= 2 w_bar sqrt(B_w' P B_w) / mu,
+% i.e. V >= (2 w_bar sqrt(B_w' P B_w) / mu)^2  := c_inv.
+%
+% This is much tighter than the isotropic ball bound — it uses the fact
+% that B_w only pokes one direction of the ellipsoid.
+P_half     = sqrtm(P);
+P_half_inv = inv(P_half);
+mu         = min(eig(P_half_inv * Qbar * P_half_inv));
+g_bound    = sqrt(B_w.' * P * B_w);      % sqrt(B_w' P B_w)
+c_inv      = (2 * w_bar * g_bound / mu)^2;
+
+fprintf('\n=== Lyapunov barrier certificate ===\n');
+fprintf('  lambda_min(P)      = %.3e\n', min(eig(P)));
+fprintf('  lambda_max(P)      = %.3e\n', max(eig(P)));
+fprintf('  sqrt(B_w'' P B_w)  = %.3e\n', g_bound);
+fprintf('  mu (Qbar eig on P-metric) = %.3e\n', mu);
+fprintf('  w_bar  (15%% P0)    = %.3e W\n', w_bar);
+fprintf('  c_inv (invariant level) = %.3e\n', c_inv);
+
+%% ===== Containment of initial set =====
+% Initial set: box around x_op with halfwidth delta_entry (matches reach_operation).
+delta_entry = [0.01 * x_op(1);
+               0.001 * abs(x_op(2:7));
+               0.1; 0.1; 0.1];
+
+% Worst-case V over the initial box: max x'Px = sum over all 2^n corners.
+% For small n we could enumerate, but the sharper bound is
+%   max V(dx) = delta_entry' * |P| * delta_entry
+% (elementwise abs of P), which is the L1 energy bound.
+c_entry = delta_entry.' * abs(P) * delta_entry;
+
+fprintf('\n  c_entry (bound on V over initial box) = %.3e\n', c_entry);
+
+gamma = max(c_entry, c_inv);   % barrier level must contain both
+fprintf('  gamma (barrier level) = %.3e\n', gamma);
+if gamma == c_entry
+    fprintf('  (initial set drives gamma — invariant piece already inside entry)\n');
+else
+    fprintf('  (disturbance inflation drives gamma)\n');
+end
+
+%% ===== Sweep Qbar(9,9) to find the tightest safe barrier =====
+% The isotropic Lyapunov bound is conservative because the "slow decay"
+% direction dominates mu even when T_c is tightly controlled.  Sweep the
+% T_c weight to find a Qbar that yields a sub-5K barrier if one exists
+% for this LQR design.
+fprintf('\n=== Sweeping Qbar(T_c) weight ===\n');
+weights = [1e1, 1e2, 1e3, 1e4, 1e5, 1e6];
+best_dTc = inf;  best_w = NaN;  best_gamma = NaN;  best_P = [];
+for wTc = weights
+    Qbar_s = Qbar;  Qbar_s(9,9) = wTc;
+    try
+        Ps = lyap(A_cl.', Qbar_s);
+    catch
+        continue
+    end
+    if any(eig(Ps) <= 0), continue, end
+    Ph = sqrtm(Ps);  Phi = inv(Ph);
+    mu_s = min(eig(Phi * Qbar_s * 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;
+    g_s_level = max(ci_s, ce_s);
+    Pinv_s = inv(Ps);
+    dTc_s = sqrt(g_s_level * (e9.' * Pinv_s * e9));
+    fprintf('  Qbar(9,9) = %.0e  ->  gamma = %.3e,  max|dT_c| = %7.3f K\n', ...
+            wTc, g_s_level, dTc_s);
+    if dTc_s < best_dTc
+        best_dTc = dTc_s;  best_w = wTc;  best_gamma = g_s_level;  best_P = Ps;
+    end
+end
+fprintf('  Best: Qbar(9,9) = %.0e  ->  max|dT_c| = %.3f K\n', best_w, best_dTc);
+if best_dTc <= delta_safe_Tc
+    fprintf('  *** TIGHT BARRIER FOUND: V(x) = x.'' P_best x - gamma ***\n');
+    P = best_P;  gamma = best_gamma;
+end
+
+%% ===== Safety: does the gamma-level set fit inside X_safe? =====
+% X_safe = { x : |T_c - T_c0| <= 5 K },  i.e. |e_9.' * dx| <= 5.
+% Max |e_9.' * dx| over {dx : dx' P dx <= gamma} is sqrt(gamma * e_9' P^-1 e_9).
+Pinv = inv(P);
+max_dTc_on_ellipsoid = sqrt(gamma * (e9.' * Pinv * e9));
+
+fprintf('\n=== Safety check on T_c ===\n');
+fprintf('  Max |dT_c| on gamma-ellipsoid = %.3f K\n', max_dTc_on_ellipsoid);
+fprintf('  Safe band                     = +/- %.1f K\n', delta_safe_Tc);
+if max_dTc_on_ellipsoid <= delta_safe_Tc
+    fprintf('  BARRIER VALID: V(x) = x.''Px - %.3e  certifies T_c safety.\n', gamma);
+else
+    fprintf('  *** BARRIER TOO LOOSE *** - ellipsoid reach into unsafe region.\n');
+    fprintf('  Try a tighter LQR (bigger Q_Tc or smaller R) or tighter X_entry.\n');
+end
+
+save(fullfile('.', 'barrier_lyapunov_result.mat'), ...
+     'P', 'gamma', 'c_entry', 'c_inv', 'w_bar', 'K', 'A_cl', 'delta_entry', ...
+     'max_dTc_on_ellipsoid', 'delta_safe_Tc', '-v7');
+
+fprintf('\nSaved barrier to ./barrier_lyapunov_result.mat\n');

reachability/reach_linear.m

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+function [T, R_lo, R_hi, center] = reach_linear(A_cl, B_w, x0_center, x0_halfwidth, w_lo, w_hi, tspan, dt)
+% REACH_LINEAR  Interval reach set for dx/dt = A_cl*x + B_w*w, w in [w_lo, w_hi].
+%
+%  Hand-rolled zonotope propagator specialized to the case where:
+%    - The initial set is an axis-aligned box around x0_center with
+%      halfwidths x0_halfwidth.
+%    - The disturbance w is a scalar interval.
+%
+%  The reach set at time t is:
+%     X(t) = {expm(A_cl*t) * x0 + integral_0^t expm(A_cl*(t-s))*B_w*w(s) ds :
+%             x0 in X0, w(s) in W}.
+%
+%  Decompose:
+%     X(t) = expm(A_cl*t) * X0  (+)  S_w(t),
+%  where S_w(t) is the reachable contribution of the disturbance.
+%  Both parts are zonotopes; we over-approximate each by a box at every
+%  time step via the matrix/integral absolute sum (interval hull).
+%
+%  Inputs:
+%    A_cl         - n x n closed-loop state matrix (Hurwitz assumed)
+%    B_w          - n x 1 disturbance vector
+%    x0_center    - n x 1 center of initial set
+%    x0_halfwidth - n x 1 halfwidths of initial set (>=0)
+%    w_lo, w_hi   - scalar disturbance bounds
+%    tspan        - [t0, tf]
+%    dt           - time step for exporting the reach tube (s)
+%
+%  Outputs:
+%    T     - M x 1 time grid
+%    R_lo  - n x M lower bounds of reach set
+%    R_hi  - n x M upper bounds of reach set
+%    center- n x M center trajectory (nominal, w = w_mid)
+
+    n = size(A_cl, 1);
+    assert(size(A_cl,2) == n, 'A_cl must be square');
+    assert(numel(x0_center) == n && numel(x0_halfwidth) == n, 'x0 dim mismatch');
+    x0_center    = x0_center(:);
+    x0_halfwidth = x0_halfwidth(:);
+    B_w          = B_w(:);
+
+    w_mid   = 0.5 * (w_lo + w_hi);
+    w_half  = 0.5 * (w_hi - w_lo);
+
+    T = (tspan(1):dt:tspan(2)).';
+    M = numel(T);
+
+    R_lo   = zeros(n, M);
+    R_hi   = zeros(n, M);
+    center = zeros(n, M);
+
+    % Disturbance contribution:
+    %   S_center(t) = int_0^t expm(A*(t-s))*B_w*w_mid ds       (signed, centerline)
+    %   S_half(t)   = int_0^t |expm(A*(t-s))*B_w| * w_half ds  (elementwise, halfwidth)
+    %
+    %  The halfwidth uses elementwise |.| because under an axis-aligned
+    %  box over-approximation, |linear-map| contributes absolute values.
+    %  This is the interval-arithmetic analog of propagating a zonotope.
+    S_center = zeros(n, 1);
+    S_half   = zeros(n, 1);
+
+    x_center_now = x0_center;
+    A_step       = expm(A_cl * dt);                 % state transition per step
+    A_abs_step   = abs(A_step);                     % for interval-halfwidth
+
+    % Integrated input gain per step:
+    %    G_step = int_0^dt expm(A*s)*B_w ds
+    % Augmented-matrix trick: upper-right block of expm([A B_w; 0 0]*dt).
+    M_aug  = expm([A_cl, B_w; zeros(1, n+1)] * dt);
+    G_step = M_aug(1:n, n+1);                       % n x 1
+    G_abs_step = abs(G_step);
+
+    halfwidth_now = x0_halfwidth;
+
+    % Initial state
+    center(:, 1) = x_center_now + S_center;
+    R_lo(:, 1)   = center(:, 1) - halfwidth_now - S_half;
+    R_hi(:, 1)   = center(:, 1) + halfwidth_now + S_half;
+
+    for k = 2:M
+        % State piece
+        x_center_now  = A_step     * x_center_now;
+        halfwidth_now = A_abs_step * halfwidth_now;
+
+        % Disturbance piece
+        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

reachability/reach_operation.m

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+%% reach_operation.m — linear reach set for operation mode (LQR closed-loop)
+%
+%  Compute a sound over-approximation of the reach set starting from a
+%  box around x_op, under LQR feedback, with Q_sg in a specified
+%  interval.  Check that T_avg stays inside the t_avg_in_range predicate
+%  for all t in [0, T_final].
+%
+%  This is the *continuous-mode obligation* for q_operation:
+%     X_entry  := { x : |x - x_op| <= delta_entry }
+%     W        := [Q_min, Q_max]
+%     X_safe   := { x : |T_c - T_c0| <= delta_safe }
+%     Obligation: ReachTube(X_entry, W, [0, T_final]) subset X_safe.
+%
+%  If this passes, we've discharged the Thrust-3 verification for one
+%  continuous mode at the level the thesis calls for.
+
+clear; clc; close all;
+addpath('../plant-model', '../plant-model/controllers');
+
+plant = pke_params();
+x_op  = pke_initial_conditions(plant);
+
+%% ===== Closed-loop linearization =====
+[A, B, B_w, ~, ~, ~] = pke_linearize(plant, x_op, 0, plant.P0);
+
+% LQR gain using the same Q, R as ctrl_operation_lqr.m.  Keep these in
+% sync if you retune there.
+Q_lqr = diag([1, 1e-3, 1e-3, 1e-3, 1e-3, 1e-3, 1e-3, 1e-2, 1e2, 1]);
+R_lqr = 1e6;
+try
+    K = lqr(A, B, Q_lqr, R_lqr);
+catch
+    [~, ~, K] = icare(A, B, Q_lqr, R_lqr);
+end
+
+A_cl = A - B*K;
+
+fprintf('\n=== Closed-loop spectrum (A - BK) ===\n');
+eigs_cl = eig(A_cl);
+fprintf('  max Re(eig) = %.3e\n', max(real(eigs_cl)));
+fprintf('  min Re(eig) = %.3e\n', min(real(eigs_cl)));
+assert(all(real(eigs_cl) < -1e-8), 'A_cl not Hurwitz — gain tuning issue');
+
+%% ===== Define sets =====
+% X_entry: 1% box on n, 0.1% boxes on precursors (their magnitudes are
+% huge due to 1/Lambda), 0.1 K on fuel, 0.1 K on coolant, 0.1 K on cold leg.
+% This represents "we entered operation mode near the steady state".
+delta_entry = [0.01 * x_op(1);                  % n
+               0.001 * abs(x_op(2:7));          % C1..C6
+               0.1;                             % T_f [C]
+               0.1;                             % T_c [C]
+               0.1];                            % T_cold [C]
+
+% Disturbance: Q_sg = [0.85*P0, 1.00*P0] -> this captures up to a 15%
+% down-step load demand (realistic load-follow envelope).
+Q_nom = plant.P0;
+Q_min = 0.85 * plant.P0;
+Q_max = 1.00 * plant.P0;
+dQ_lo = Q_min - Q_nom;     % -0.15 * P0
+dQ_hi = Q_max - Q_nom;     % 0
+
+% X_safe: |T_c - T_c0| <= 5 K (roughly matching the "t_avg_in_range"
+% predicate window — adjust once the FRET predicate threshold is locked).
+delta_safe_Tc = 5.0;   % [C]
+
+%% ===== Reach set =====
+% Propagate in deviation coordinates: dx = x - x_op.
+tspan = [0, 600];
+dt    = 0.5;
+
+[T, R_lo, R_hi, C] = reach_linear(A_cl, B_w, zeros(10,1), delta_entry, dQ_lo, dQ_hi, tspan, dt);
+
+% Translate back to absolute coordinates for reporting
+Xabs_lo = R_lo + x_op;
+Xabs_hi = R_hi + x_op;
+Cabs    = C    + x_op;
+
+%% ===== Safety check =====
+T_c_lo = Xabs_lo(9, :);
+T_c_hi = Xabs_hi(9, :);
+
+violation_mask = (T_c_hi > plant.T_c0 + delta_safe_Tc) | ...
+                 (T_c_lo < plant.T_c0 - delta_safe_Tc);
+fprintf('\n=== Operation-mode reach-set safety ===\n');
+fprintf('  Horizon               = [%g, %g] s\n', tspan(1), tspan(2));
+fprintf('  Entry box T_c  [C]   = [%.3f, %.3f]  (x_op +/- %.1f C)\n', ...
+        x_op(9) - delta_entry(9), x_op(9) + delta_entry(9), delta_entry(9));
+fprintf('  Disturbance Q_sg     = [%.3f, %.3f] MW\n', Q_min/1e6, Q_max/1e6);
+fprintf('  Safe band on T_c     = x_op(T_c0) +/- %.1f C  -> [%.3f, %.3f]\n', ...
+        delta_safe_Tc, plant.T_c0 - delta_safe_Tc, plant.T_c0 + delta_safe_Tc);
+fprintf('  Reach T_c envelope   = [%.3f, %.3f]\n', min(T_c_lo), max(T_c_hi));
+if any(violation_mask)
+    t_first = T(find(violation_mask, 1));
+    fprintf('  *** SAFETY VIOLATED at t = %.2f s ***\n', t_first);
+else
+    fprintf('  OK: reach set stays inside the safe band.\n');
+end
+
+%% Per-state reach-set growth diagnostic (final time vs initial)
+state_names = {'n','C1','C2','C3','C4','C5','C6','T_f','T_c','T_cold'};
+fprintf('\n=== Reach-set width at t=0 vs t=T_final ===\n');
+fprintf('  %-7s  %-14s  %-14s  %-8s\n', 'state', 'init halfwidth', 'final halfwidth', 'ratio');
+for i = 1:10
+    hi = 0.5 * (R_hi(i, 1)   - R_lo(i, 1));
+    hf = 0.5 * (R_hi(i, end) - R_lo(i, end));
+    fprintf('  %-7s  %-14.4e  %-14.4e  %-8.2f\n', state_names{i}, hi, hf, hf/max(hi,eps));
+end
+
+%% ===== Plots =====
+figdir = fullfile('..', 'docs', 'figures');
+if ~exist(figdir, 'dir'), mkdir(figdir); end
+CtoF = @(T) T*9/5 + 32;
+
+% Two-panel plot: wide view with safety band, zoom view showing actual tube.
+figure('Position', [100 80 1400 500], 'Name', 'Reach tube: T_c');
+
+subplot(1,2,1);
+fill([T; flipud(T)], CtoF([T_c_hi.'; flipud(T_c_lo.')]), [1.0 0.85 0.85], ...
+     'EdgeColor', 'none'); hold on;
+plot(T, CtoF(Cabs(9, :)), 'r-',  'LineWidth', 1.2);
+yline(CtoF(plant.T_c0 + delta_safe_Tc), 'k--', 'LineWidth', 1.0);
+yline(CtoF(plant.T_c0 - delta_safe_Tc), 'k--', 'LineWidth', 1.0);
+yline(CtoF(plant.T_c0),                 'k:',  'LineWidth', 1.0);
+xlabel('Time [s]'); ylabel('T_{avg} [F]'); grid on;
+title('Safety-band view');
+legend('reach tube', 'nominal', 'safety +/- 5 C', 'Location', 'best');
+
+subplot(1,2,2);
+Tc_dev_lo = T_c_lo.' - plant.T_c0;    % M x 1, deviation in K
+Tc_dev_hi = T_c_hi.' - plant.T_c0;
+fill([T; flipud(T)], [Tc_dev_hi; flipud(Tc_dev_lo)], [1.0 0.85 0.85], ...
+     'EdgeColor', 'none'); hold on;
+plot(T, Cabs(9, :).' - plant.T_c0, 'r-', 'LineWidth', 1.2);
+yline(0, 'k:', 'LineWidth', 1.0);
+xlabel('Time [s]'); ylabel('T_{avg} - T_{c0} [K]'); grid on;
+max_dev = max(abs([Tc_dev_lo; Tc_dev_hi]));
+title(sprintf('Zoomed: max |dT_c| = %.3e K', max_dev));
+
+sgtitle(sprintf('Operation-mode reach tube, LQR, Q_{sg} in [%.0f%%, %.0f%%] P_0', ...
+                100*Q_min/Q_nom, 100*Q_max/Q_nom));
+exportgraphics(gcf, fullfile(figdir, 'reach_operation_Tc.png'), 'Resolution', 150);
+
+figure('Position', [100 80 1100 500], 'Name', 'Reach tube: n');
+fill([T; flipud(T)], [R_hi(1,:).' + x_op(1); flipud(R_lo(1,:).' + x_op(1))], ...
+     [0.85 0.85 1.0], 'EdgeColor', 'none'); hold on;
+plot(T, Cabs(1, :), 'b-', 'LineWidth', 1.2);
+xlabel('Time [s]'); ylabel('n'); grid on;
+title('Operation mode reach tube on normalized power');
+legend('reach tube', 'nominal', 'Location', 'best');
+exportgraphics(gcf, fullfile(figdir, 'reach_operation_n.png'), 'Resolution', 150);
+
+save(fullfile('.', 'reach_operation_result.mat'), ...
+     'T', 'R_lo', 'R_hi', 'C', 'Xabs_lo', 'Xabs_hi', 'Cabs', ...
+     'K', 'A_cl', 'x_op', 'delta_entry', 'Q_min', 'Q_max', 'delta_safe_Tc', '-v7');
+
+fprintf('\nSaved reach result to ./reach_operation_result.mat\n');