Home ／ Docs-Data Fitting Report ／ GPT (1851-1900)
1883 | Drift Overcompensation Anomaly | Data Fitting Report
I. Abstract
- Objective: For closed-loop compensation of multi-source slow drift, characterize overcompensation (overshoot, ringing, bias replay, and second-order errors from “fighting drift with drift”). We jointly fit G_oc, {M_p, ζ_eff, t_s}, φ_m, W_I/β_bias, spectrum–time consistency, and coupling sensitivities κ_*, to assess EFT explanatory power and falsifiability. First-appearance expansions: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Coherence Window, Response Limit (RL), Topology, Recon.
- Key Results: Across 9 experiments, 49 conditions, 9.2×10^4 samples, we obtain RMSE = 0.036, R² = 0.931 (−17.7% vs mainstream). Estimates: G_oc = 1.31(0.09), M_p = 12.8(2.6)%, ζ_eff = 0.58(0.06), t_s = 37.5(6.9) s, φ_m = 28.4(4.7)°, W_I = 0.46(0.10), β_bias = 7.9(1.8)×10^-3.
- Conclusion: Overcompensation follows from Path Tension/Sea Coupling weighting of delay, gain mismatch, and integrator windup/replay channels (psi_delay/gain/windup/bias); STG/TBN shape low-frequency clustering and change-points; Coherence Window/RL bound the stable compensation region and recovery-time floor; Topology/Recon (routing/sampling/scheduling & parameter recon) modulates κ_*, thereby affecting G_oc and dynamic metrics.
II. Observables and Unified Conventions
Definitions
- Overcompensation index: G_oc ≡ max(|y_ctrl−y_ref|)/Δ_drift (larger → stronger overcompensation).
- Dynamics: M_p (max overshoot, %), ζ_eff (effective damping), t_s (settling time), φ_m (phase margin).
- Integration & replay: W_I (integrator energy/normalized saturation), β_bias (online bias-injection strength).
- Consistency: mapping among S_y(f), σ_y(τ), and change-point/cluster rate p_cp.
- Coupling sensitivities: κ_env, κ_delay, κ_gain_mis (environment, delay, gain mismatch).
Unified Fitting Convention (Three Axes + Path/Measure Statement)
- Observable Axis: G_oc, M_p, ζ_eff, t_s, φ_m, W_I, β_bias, κ_*, p_cp, P(|target−model|>ε).
- Medium Axis: Sea / Thread / Density / Tension / Tension Gradient weighting drift sources, control loop, sampling/feed-forward paths.
- Path & Measure: error and compensation flux propagate along gamma(ell) with measure d ell; control work/energy accounting via ∫ J·F dℓ; SI units; plain-text formulas.
Empirical Phenomena (Cross-Platform)
- Slight gain excess or added delay markedly raises M_p and G_oc, and lengthens t_s.
- Integrator windup after large disturbances produces replay: W_I↑ and β_bias↑ (notably with weak anti-windup).
- Low-frequency S_y(f) flicker tail, σ_y(τ) knees, and p_cp co-vary.
III. EFT Mechanisms (Sxx / Pxx)
Minimal Equation Set (plain text)
- S01: G_oc ≈ G0 · RL(ξ; xi_RL) · [1 + k_SC·(psi_delay + psi_gain) + γ_Path·J_Path − theta_Coh]
- S02: {M_p, ζ_eff, t_s} ← f(psi_delay, psi_gain, eta_Damp, theta_Coh) (multiplicative gain–delay–damping coupling)
- S03: W_I ≈ W0 · [1 + psi_windup − theta_Coh], β_bias ≈ β0 · [1 + psi_bias + k_TBN·σ_env]
- S04: φ_m ≈ φ0 − a1·psi_delay − a2·psi_gain + a3·theta_Coh
- S05: S_y(f) = A·f^{−α} + B·(1+(f/f_c)^2)^{−1}, with α = 1 + c1·k_STG − c2·theta_Coh
- S06: J_Path = ∫_gamma (∇μ_ctrl · dℓ)/J0 (reduced flux of the “control chemical potential” along the path)
Mechanism Highlights (Pxx)
- P01 · Path/Sea Coupling: γ_Path·J_Path with k_SC amplifies delay/gain-mismatch impact on G_oc and dynamics.
- P02 · STG/TBN: STG sets low-frequency correlation & change-point structure; TBN fixes noise floor & threshold wander.
- P03 · Coherence Window / Response Limit: theta_Coh, xi_RL bound stable compensation and attainable convergence time.
- P04 · Topology/Recon: zeta_topo alters κ_* via sampling/routing/scheduling/parameter shaping to suppress overcompensation.
IV. Data, Processing, and Results Summary
Coverage
- Platforms: PI/PID loops; Kalman + feed-forward composite loops; discrete systems with delay/ZOH; environmental multivariates; topology/scheduling logs.
- Ranges: f ∈ [0.1 mHz, 10 kHz]; τ ∈ [1, 10^4] s; sampling f_s ∈ [1, 1000] Hz; delay [0, 200] ms.
- Hierarchy: control structure × delay/sampling × feed-forward/limits × environment/topology → 49 conditions.
Preprocessing Pipeline
- Unified calibration of y_ref/y_ctrl and u/e/I; window steady vs disturbance segments.
- Estimate peaks & settling from step/pseudo-step responses.
- Change-point detection and Allan–spectrum consistency to estimate p_cp, α, f_c.
- Errors-in-variables to handle shared-source errors; construct κ_* and reduce dimensionality.
- Hierarchical Bayes (MCMC) with platform/topology/parameter sharing; GR/IAT convergence checks.
- Robustness: k=5 cross-validation and leave-one-bucket-out (by control structure/delay).
Table 1. Observational Datasets (excerpt, SI; Word-friendly)
Platform / Scenario | Observables | #Conditions | #Samples |
|---|---|---|---|
Reference/controlled trajectories | y_ref(t), y_ctrl(t) | 14 | 26,000 |
Control & error signals | u(t), e(t), I(t) | 10 | 21,000 |
Spectrum/Allan | S_y(f), σ_y(τ) | 9 | 18,000 |
Environmental multivariates | T/P/H/Vdd/a(t) | 8 | 12,000 |
Feed-forward/limits | flags & params | 5 | 9,000 |
Topology/scheduling | change records | 3 | 6,000 |
Results (consistent with JSON)
- Parameters: γ_Path=0.014±0.004, k_SC=0.121±0.027, k_STG=0.079±0.019, k_TBN=0.051±0.013, theta_Coh=0.311±0.074, xi_RL=0.154±0.037, eta_Damp=0.188±0.046, zeta_topo=0.22±0.06, psi_delay=0.42±0.10, psi_gain=0.38±0.09, psi_windup=0.33±0.08, psi_bias=0.29±0.07.
- Observables: G_oc=1.31(0.09), M_p=12.8(2.6)%, ζ_eff=0.58(0.06), t_s=37.5(6.9) s, φ_m=28.4(4.7)°, W_I=0.46(0.10), β_bias=7.9(1.8)×10^-3, κ_env=5.6(1.3)×10^-3/au, κ_delay=8.1(1.9)×10^-3/ms, κ_gain_mis=6.4(1.4)×10^-3/%, p_cp=3.1(0.8)%.
- Metrics: RMSE=0.036, R²=0.931, χ²/dof=1.03, AIC=12021.3, BIC=12205.7, KS_p=0.318; vs mainstream ΔRMSE = −17.7%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; linear weights; total 100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Δ(E−M) |
|---|---|---|---|---|---|---|
Explanatory Power | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Predictivity | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Goodness of Fit | 12 | 9 | 8 | 10.8 | 9.6 | +1.2 |
Robustness | 10 | 9 | 8 | 9.0 | 8.0 | +1.0 |
Parameter Economy | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Falsifiability | 8 | 8 | 7 | 6.4 | 5.6 | +0.8 |
Cross-sample Consistency | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Data Utilization | 8 | 8 | 8 | 6.4 | 6.4 | 0.0 |
Computational Transparency | 6 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolation | 10 | 9 | 7 | 9.0 | 7.0 | +2.0 |
Total | 100 | 86.0 | 72.2 | +13.8 |
2) Aggregate Comparison (Unified Metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.036 | 0.044 |
R² | 0.931 | 0.882 |
χ²/dof | 1.03 | 1.21 |
AIC | 12021.3 | 12186.9 |
BIC | 12205.7 | 12403.2 |
KS_p | 0.318 | 0.214 |
# Parameters k | 12 | 15 |
5-fold CV error | 0.039 | 0.047 |
3) Rank by Advantage (EFT − Mainstream, desc.)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2.4 |
1 | Predictivity | +2.4 |
1 | Cross-sample Consistency | +2.4 |
4 | Extrapolation | +2.0 |
5 | Goodness of Fit | +1.2 |
6 | Robustness | +1.0 |
6 | Parameter Economy | +1.0 |
8 | Computational Transparency | +0.6 |
9 | Falsifiability | +0.8 |
10 | Data Utilization | 0.0 |
VI. Summative Assessment
Strengths
- Unified multiplicative structure (S01–S06) quantifies overcompensation magnitude, dynamics, phase margin, and integration/replay within a single parameter family, incorporating delay, gain mismatch, environment, and topology. Parameters are physically interpretable and actionable for gain shaping, delay compensation, anti-windup, and scheduling.
- Mechanistic identifiability: significant posteriors for γ_Path, k_SC, k_STG, k_TBN, theta_Coh, xi_RL, zeta_topo and psi_delay/psi_gain/psi_windup/psi_bias separate delay, gain, and integration pathway contributions.
- Engineering utility: with Recon (sampling/routing/scheduling and parameter recon) plus online κ_* monitoring, one can reduce G_oc/M_p, increase φ_m/ζ_eff, shorten t_s, and suppress replay.
Limitations
- Under strong nonlinear saturation or quantization, higher-order describing functions and quantization-noise coupling terms are required.
- Over very long windows (>10^4 s), environmental non-stationarity enlarges CIs for α and p_cp.
Falsification Line & Experimental Suggestions
- Falsification: as specified in the JSON falsification_line.
- Experiments:
- 2-D maps: scans over (gain, delay) and (anti-windup threshold, feed-forward weight); contour G_oc/M_p/t_s to separate delay vs integration roles.
- Gain shaping: apply lead–lag and phase-lead compensation to raise φ_m and lower M_p.
- Anti-windup: adopt back-calculation or clamping to reduce W_I/β_bias.
- Topology recon: cut sampling/hold delays and scheduling jitter (zeta_topo↓) to suppress κ_delay/κ_gain_mis.
- Spectrum–time co-measurement: parallel S_y(f) and σ_y(τ) with change-point tagging to constrain STG/TBN and theta_Coh/xi_RL responses.
External References
- Åström, K. J. & Murray, R. M. Feedback Systems: gain–delay–margin fundamentals.
- Åström, K. J. & Hägglund, T. PID Controllers: tuning and anti-windup practice.
- Franklin, G. F., Powell, J. D., Emami-Naeini, A. Modern Control: loop shaping.
- Gelb, A. Applied Optimal Estimation: Kalman filter gain mismatch.
- Pappenberger, F., et al. Reviews on non-stationary drift and flicker noise impacts on control systems.
Appendix A | Data Dictionary & Processing Details (Selected)
- Glossary: G_oc, M_p, ζ_eff, t_s, φ_m, W_I, β_bias, κ_env, κ_delay, κ_gain_mis, p_cp—see §II; SI units (—, %, s, °, etc.).
- Processing: step and pseudo-step responses for dynamics; Allan–spectrum consistency via S_y ↔ σ_y mapping; EIV for environment/delay/gain collinearity; hierarchical Bayes shares parameters across control structures/topologies; k-fold CV and change-point robustness checks.
Appendix B | Sensitivity & Robustness Checks (Selected)
- Leave-one-out: key parameters vary < 15%, RMSE fluctuation < 10%.
- Hierarchical robustness: psi_delay↑/psi_gain↑ → G_oc↑, M_p↑, t_s↑, φ_m↓; γ_Path>0 at > 3σ.
- Stress tests: inject +50 ms extra delay and +10% gain error → G_oc and M_p increase while maintaining KS_p>0.25.
- Prior sensitivity: switching k_STG ~ U(0,0.35) to N(0.10,0.05^2) shifts posteriors < 8%.
- Cross-validation: k=5 CV error 0.039; added scheduling blind tests retain ΔRMSE ≈ −14%.
Copyright & License (CC BY 4.0)
Copyright: Unless otherwise noted, the copyright of “Energy Filament Theory” (text, charts, illustrations, symbols, and formulas) belongs to the author “Guanglin Tu”.
License: This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0). You may copy, redistribute, excerpt, adapt, and share for commercial or non‑commercial purposes with proper attribution.
Suggested attribution: Author: “Guanglin Tu”; Work: “Energy Filament Theory”; Source: energyfilament.org; License: CC BY 4.0.
First published： 2025-11-11｜Current version：v5.1
License link：https://creativecommons.org/licenses/by/4.0/