Dane Sabo 02a675c152 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>

2026-04-17 12:52:37 -04:00

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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:

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).

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