dc4cfed61a
Per Dane's question: does LQR actually factor into the 2364x barrier
on n_high_trip, or is that just open-loop plant?
Answer: LQR IS included (A_cl = A - B*K), and the open-loop version is
catastrophically worse. Results on inv2_holds halfspaces:
open-loop LQR closed-loop
fuel_centerline 26.9M K bound 1137 K bound
t_avg_high_trip 788220 K bound 33.2 K bound
n_high_trip 27.4M x bound 1242 x bound
cold_leg_subcooled 1.8M K bound 77.8 K bound
gamma (level) 1.04e13 1.85e4
LQR improves every bound by ~20,000x — dramatic help — but the bounds
are still physically meaningless. The ceiling is set by plant anisotropy
(Lambda=1e-4 vs thermal timescales ~ seconds) forcing P to be
ill-conditioned regardless of LQR tuning. mu (slowest V-decay rate)
barely moves between OL and CL because both share the same slowest
thermal mode.
Clean motivation for the thesis chapter's move to polytopic / SOS
barriers: quadratic Lyapunov hits an anisotropy ceiling that no amount
of controller work can fix.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
87 lines
3.4 KiB
Matlab
87 lines
3.4 KiB
Matlab
%% barrier_compare_OL_CL.m — does LQR help the Lyapunov barrier, or not?
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%
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% Head-to-head: solve the same Lyapunov-ellipsoid barrier analysis on
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% (i) the open-loop plant A (u = 0)
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% (ii) the LQR closed loop A_cl = A - B*K
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% with identical Qbar and identical disturbance bound. Report per-
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% halfspace bounds against inv2_holds so we can see exactly where LQR
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% helps and where it doesn't.
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%
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% The plant is already Hurwitz open-loop (max Re(eig) = -0.012 /s), so
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% the Lyapunov equation has a solution in both cases.
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clear; clc;
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addpath('../plant-model', '../plant-model/controllers');
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plant = pke_params();
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x_op = pke_initial_conditions(plant);
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pred = load_predicates(plant);
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[A, B, B_w, ~, ~, ~] = pke_linearize(plant, x_op, 0, plant.P0);
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%% ===== LQR gain =====
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Q_lqr = diag([1, 1e-3, 1e-3, 1e-3, 1e-3, 1e-3, 1e-3, 1e-2, 1e2, 1]);
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R_lqr = 1e6;
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try
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K = lqr(A, B, Q_lqr, R_lqr);
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catch
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[~, ~, K] = icare(A, B, Q_lqr, R_lqr);
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end
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A_cl = A - B*K;
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%% ===== Shared pieces =====
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Qbar = diag([1, 1e-4, 1e-4, 1e-4, 1e-4, 1e-4, 1e-4, 1, 1e2, 1]); % best from the earlier sweep
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w_bar = 0.15 * plant.P0;
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delta_entry = [0.01 * x_op(1);
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0.001 * abs(x_op(2:7));
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0.1; 0.1; 0.1];
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inv2 = pred.mode_invariants.inv2_holds;
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A_inv = inv2.A_poly; b_inv = inv2.b_poly; comps = inv2.components;
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b_dev = b_inv - A_inv * x_op; % headroom in deviation coords
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%% ===== Helper =====
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function [gamma, maxima, eig_cl] = run_case(Acase, B_w, Qbar, w_bar, delta_entry, A_inv, b_dev)
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% Solve Lyapunov, compute invariant level, per-halfspace maxima.
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P = lyap(Acase.', Qbar);
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Ph = sqrtm(P); Phi = inv(Ph);
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mu = min(eig(Phi * Qbar * Phi));
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g_bound = sqrt(B_w.' * P * B_w);
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c_inv = (2 * w_bar * g_bound / mu)^2;
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c_entry = delta_entry.' * abs(P) * delta_entry;
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gamma = max(c_inv, c_entry);
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Pinv = inv(P);
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maxima = zeros(size(A_inv, 1), 1);
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for k = 1:size(A_inv, 1)
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a = A_inv(k, :).';
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maxima(k) = sqrt(gamma * (a.' * Pinv * a));
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end
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eig_cl = eig(Acase);
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end
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%% ===== Run both =====
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[gamma_OL, max_OL, eig_OL] = run_case(A, B_w, Qbar, w_bar, delta_entry, A_inv, b_dev);
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[gamma_CL, max_CL, eig_CL] = run_case(A_cl, B_w, Qbar, w_bar, delta_entry, A_inv, b_dev);
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%% ===== Report =====
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fprintf('\n=== Open-loop vs LQR closed-loop Lyapunov barrier ===\n');
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fprintf(' max Re(eig) OL = %.3e CL = %.3e\n', max(real(eig_OL)), max(real(eig_CL)));
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fprintf(' min Re(eig) OL = %.3e CL = %.3e\n', min(real(eig_OL)), min(real(eig_CL)));
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fprintf(' gamma OL = %.3e CL = %.3e (ratio CL/OL = %.3g)\n', ...
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gamma_OL, gamma_CL, gamma_CL/gamma_OL);
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fprintf('\n Per-halfspace max |a'' dx| on gamma-ellipsoid:\n');
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fprintf(' %-22s %-10s %-12s %-12s %-12s %-10s\n', ...
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'halfspace', 'headroom', 'open-loop', 'closed-loop', 'CL - OL', 'ratio');
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for k = 1:size(A_inv, 1)
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ratio = max_CL(k) / max_OL(k);
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fprintf(' %-22s %10.3f %12.3f %12.3f %+12.3f %8.3fx\n', ...
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comps{k}, b_dev(k), max_OL(k), max_CL(k), max_CL(k) - max_OL(k), ratio);
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end
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fprintf('\n Interpretation:\n');
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fprintf(' - CL < OL on a row ==> LQR tightens that halfspace.\n');
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fprintf(' - CL ~= OL ==> LQR does not help that direction at all.\n');
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fprintf(' - CL > OL ==> LQR made that direction WORSE for the barrier (!!).\n');
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