{
 "artifacts": null,
 "category": "dynamics",
 "description": "Add a low-dimensional feedback correction to the neural reference so that accumulated position mismatch is removed when actuator saturation or kinematic mismatch causes the shaped trajectory to lag the requested one. Unlike ordinary integral action, the correction is passed through the same feasibility-preserving reference shaper, preventing integral windup while ensuring that compensation cannot violate current, voltage, speed, or acceleration limits.",
 "download_zip": "https://synthcore.org/download/1127",
 "formulas_latex": [
  "$$c_{t+1}=\\rho c_t+K_I e_t+K_A\\bigl(r_t-\\tilde z_t\\bigr),\\qquad \\tilde z_t=z_t+K_Pe_t+K_Cc_t,$$",
  "$$r_t=\\mathcal{S}(x_t,\\tilde z_t)=\\arg\\min_r\\frac{1}{2}(r-\\tilde z_t)^\\top W(r-\\tilde z_t)\\quad\\mathrm{s.t.}\\quad g_j(r,x_t)\\leq0,$$",
  "$$\\rho_{\\mathrm{spec}}\\!\\left(\\begin{bmatrix}A\u0026-BK_C\\\\K_I C\u0026\\rho I\\end{bmatrix}\\right)\u003c1,$$"
 ],
 "id": 2907,
 "implementation": "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.",
 "math_summary": "Let $e_t=y^{\\mathrm{des}}_t-y_t$ be the position or task-space error, $c_t$ the compensator state, and $z_t$ the neural reference. Use an accumulated-error update $c_{t+1}=\\rho c_t+K_Ie_t$, where $0\\leq\\rho\\leq1$ is leakage and $K_I$ is an integral gain matrix. Form the compensated command $\\tilde z_t=z_t+K_Cc_t+K_Pe_t$, then apply the constrained shaping map $r_t=\\mathcal{S}(x_t,\\tilde z_t)$. To prevent windup, integrate the shaping residual $d_t=r_t-\\tilde z_t$ through an anti-windup gain $K_A$: $c_{t+1}=\\rho c_t+K_Ie_t+K_A d_t$. For local linear error dynamics $e_{t+1}=Ae_t-BK_Cc_t+\\delta_t$, a sufficient unsaturated design target is that the augmented closed-loop matrix have spectral radius below one. Saturation changes the attainable equilibrium but cannot violate hard constraints because $r_t$ is always shaped.",
 "math_tags": [
  "control-theory",
  "dynamical-systems",
  "linear-algebra",
  "optimization"
 ],
 "ml_areas": [
  "rl",
  "training-dynamics",
  "world-model"
 ],
 "paper": {
  "arxiv_id": "2608.30825",
  "arxiv_url": "https://arxiv.org/abs/2608.30825",
  "summary_what_math_gives_to_ml": "The paper offers a constructive real-time optimization mechanism for nonlinear, state-dependent actuator constraints: KKT conditions reduce the global reference-shaping problem to a finite set of candidate points, which can be evaluated deterministically using closed-form expressions and a small eigenvalue problem. This is transferable as a neural-network output shield or reference layer that converts an unconstrained policy prediction into the closest feasible command without iterative projection or backpropagated optimization. The accompanying compensator suggests a second mechanism for correcting accumulated position mismatch caused by aggressive but feasible commands while retaining constraint feasibility. The strongest near-term test is whether the finite-candidate layer maintains feasibility at high control rates and exhibits a predictable transition from exact tracking to constrained tracking as the requested NN command crosses the feasible boundary.",
  "title": "Real-Time Reference Shaping for Servo Systems",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 5,
  "usefulness": 7
 },
 "solves": [
  "accuracy",
  "stability"
 ],
 "title": "Feasibility-Preserving Error Compensator",
 "url": "https://synthcore.org/idea/2907/feasibility-preserving-error-compensator",
 "verification": {
  "peer_reviewed": false,
  "status": "queued_mechanism",
  "status_label": "Queued — mechanism check",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}