PWR-HYBRID-3/reachability

Reachability

Continuous-mode verification for the PWR_HYBRID_3 hybrid controller.

Soundness status: APPROXIMATE

The current reach_operation.m result is not a sound reach tube for the physical plant. It is a sound over-approximation of the linearized closed-loop system (A_cl = A - BK around x_op) under bounded disturbance. The linear model is itself an approximation of the nonlinear plant (../plant-model/pke_th_rhs.m), and that approximation error is not currently bounded or inflated into the tube.

Two paths to upgrade to a sound result:

1. Nonlinear reach directly — CORA nonlinearSys, JuliaReach BlackBoxContinuousSystem, or equivalent. More expensive but the honest answer.
2. Linear reach + Taylor-remainder inflation — compute an upper bound on ||f_nl(x, u) - (A x + B u)|| over the reach set (via Hessian norm estimate on each component of f_nl) and inflate the linear tube by that bound. Less expensive, still rigorous.

Both are thesis-blocking for any safety claim. Deferred only until the per-mode plumbing is solid; it is not a "nice to have".

The current 5-orders-of-margin buffer (reach envelope ~0.03 K against a 5 K safety band) means linearization error would have to be huge to invalidate the conclusion, but that is vibes, not a proof.

Related open issues

• Saturation semantics. ctrl_heatup.m uses sat(u, u_min, u_max). Saturation is formally a 3-mode piecewise-affine system. For heatup reach this has to be handled as (a) hybrid locations, or (b) proven dormant via reach on u_unsat. Not modeled in the current artifacts (operation-mode LQR saturation is dormant in practice but the proof is implicit).
• Parametric uncertainty in α_f, α_c. Real plants have α drift with burnup (~20%), boron (α_c ranges 10×), xenon. The feedback-linearization in ctrl_heatup.m assumes exact α; a robust treatment would make α an interval and propagate parametric reach. Currently idealized — flag in the chapter.

What's here

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

• Any sound reach tube (see top of this file).
• Nonlinear reach for the original P controller on operation.
• Heatup reach (ramped reference makes x* time-varying — needs trajectory-LQR or a different formulation, and the saturation semantics need to be made explicit).
• Shutdown, scram, initialization reach.
• Hybrid-system level verification (mode switching validity).
• Parametric robustness to α_f, α_c drift.