Feasibility-Preserving Error Compensator

1. Integration point: use this module at inference time in a servoing policy, learned trajectory generator, or world-model controller. The NN predicts nominal reference $z_t$; the compensator uses measured output $y_t$ and desired output $y_t^{\mathrm{des}}$ before the finite-candidate shaper. Keep $c_t$ separate from the NN hidden state so it can be reset between episodes. 2. Pseudocode: initialize $c=0$; at each control tick compute $e=y_{\mathrm{des}}-y$, $\tilde z=z+K_Pe+K_Cc$; call the candidate shaper to obtain $r$; compute $d=r-\tilde z$; update $c\leftarrow\rho c+K_Ie+K_A d$; send $r$ to the low-level controller. Clip $c$ only as a numerical safeguard. 3. Computed versus estimated: $e$, $d$, and shaped references are directly computed; local matrices $A$, $B$, and $C$ are estimated by finite differences or system identification. Tune gains from the linearized model, reducing $K_I$ until the predicted augmented spectral radius is below 0.9. 4. First cheap experiment: use a small MLP to command a simulated servo tracking square-wave and sinusoidal references under tight voltage and current limits. Compare no compensation, ordinary integral action before saturation, and this residual anti-windup method. Measure RMS position error, recovery time, maximum constraint violation, and compensator norm. The prediction is zero feasibility violations with reduced post-saturation error; in unsaturated tests, measured error decay should follow the dominant eigenvalue of the augmented matrix within 20%. Ordinary integral action should show larger windup and delayed recovery when saturation persists.