02a675c152
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>
94 lines
3.4 KiB
Matlab
94 lines
3.4 KiB
Matlab
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
|