21-EFT.WP.Metrology.Sync v1.0 | Chapter 6 — Servo Modeling & Filtering (PLL / FLL / PI / PII / Kalman)

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Chapter 6 — Servo Modeling & Filtering (PLL / FLL / PI / PII / Kalman)

One-line objective: Build a unified clock-servo and filtering framework—deriving computable equations, tuning, and robustness strategies for PLL/FLL/PI/PII and Kalman—so that under diverse protocols and noise conditions the loop converges stably and meets SLOs on offset/skew/J.

I. Scope & Objects

1. Applicable targets
   • End-to-end time/phase/frequency servos: local clock closed-loop to upstream references (PTP/NTP/WR/GNSS/SyncE).
   • Operating modes: offline replay tuning, online adaptation, multi-domain streaming.
2. Inputs
   • Timestamps & measurements: y_k (observed offset, seconds), Delta_k (RTT/OWD estimate), t_k (sample time, step Ts).
   • Noise: R (measurement), Q (process), sigma_ADEV(τ) (see TimeBase Ch. 7).
   • Protocol metadata: hwts, profile_id, correctionField, asym, dual-form arrival delta_form.
3. Outputs
   • Estimates & control: hat{phi}_k (phase/time error), hat{f}_k (frequency error, ppb), u_k (control), state ∈ {free, acquire, track, holdover}.
   • Quality & compliance: offset_p95/p99, tdev(τ), MTIE(τ), contract evaluation, and manifest.sync.servo.*.

II. Terms & Variables

1. States & observations
   • phi_k: local vs. reference phase/time error (s)
   • f_k: fractional frequency error (ppb)
   • a_k: frequency drift (ppb/s, optional third order)
   • y_k: observed offset (post-protocol compensation)
   • Ts: sampling period (s)
2. Control & gains
   • u_k: control actuation (incremental frequency correction)
   • Kp, Ki, Kii: PI/PII gains
   • B: loop bandwidth (Hz), zeta: damping, wn: natural frequency (rad/s)
3. Noise & filtering
   • Q, R, P_k: Kalman process/measurement and error covariances
   • j_gate: gating statistic (chi-square or robust distance)
   • psi: stability index (see TimeBase & CrossStats)

*III. Axioms P606- **

• P606-1 (Frequency-before-phase): Establish frequency alignment (FLL/SyncE/NIC PLL) before phase/time lock (PLL/KF).
• P606-2 (Interchangeable noise families): Decompose noise per Allan taxonomy—white PM / flicker PM / white FM / random-walk FM—and design loop bandwidth to satisfy target tdev(τ).
• P606-3 (Dimensional integrity): check_dim(expr) everywhere; unit(phi)="s", unit(f)="ppb", unit(u)="ppb".
• P606-4 (Dual-form logging): When arrival/link estimates correct observations, record both T_arr forms and delta_form.
• P606-5 (Robust-first): Prefer gating and robust costs (Huber/Tukey) over hard rejection; on gating failure, enter holdover.
• P606-6 (Traceable tuning): Persist the origins, derivations, and versions of B, zeta, Q, R in the manifest.

*IV. Minimal Equations S606- **

• S606-1 (Discrete second-order clock model)
  x_k = [ phi_k, f_k ]^T
  x_{k+1} = A x_k + B u_k + w_k，y_k = H x_k + v_k
  with A = [[1, Ts],[0, 1]], B = [[Ts],[1*0]] (control on phase via frequency integral) or B = [[Ts],[0]] (direct to phase increment), and H = [1, 0]。
• S606-2 (PI/PII control laws)
  PI: u_k = - Kp * y_k - Ki * s_k，s_k = s_{k-1} + y_k
  PII: u_k = - Kp * y_k - Ki * s_k - Kii * r_k，r_k = r_{k-1} + s_k
  Closed-loop difference (PI): phi_{k+1} = phi_k + Ts * f_k + Ts * u_k，f_{k+1} = f_k。
• S606-3 (FLL & PLL–FLL composite)
  Frequency observation: y_fll,k = ( y_k - y_{k-1} ) / Ts
  FLL: u_k = - Kf * y_fll,k；Composite: u_k = - Kp * y_k - Ki * s_k - Kf * y_fll,k。
• S606-4 (Kalman servo, second order)
  Predict: x_{k|k-1} = A x_{k-1|k-1} + B u_{k-1}，P_{k|k-1} = A P_{k-1|k-1} A^T + Q
  Update: K_k = P_{k|k-1} H^T ( H P_{k|k-1} H^T + R )^{-1}
  x_{k|k} = x_{k|k-1} + K_k ( y_k - H x_{k|k-1} )，P_{k|k} = ( I - K_k H ) P_{k|k-1}
  Control: u_k = - G * [ y_k, hat{f}_k ]^T (design matrix G; degenerate to PI when needed)。
• S606-5 (Third-order model with drift)
  x_k = [ phi_k, f_k, a_k ]^T，A = [[1, Ts, 0.5*Ts^2],[0,1,Ts],[0,0,1]]，H = [1,0,0] — suited for long holdover/aging.
• S606-6 (Bandwidth/damping → gains, approximation)
  With wn = 2 * pi * B, bilinear digital approximation: Kp ≈ 2 * zeta * wn * Ts，Ki ≈ ( wn^2 ) * Ts^2。
  For PII: choose Kii so the three closed-loop poles cluster near {e^{-wn Ts}, e^{-wn Ts}, e^{-wn Ts}} (via pole placement or numeric solve).
• S606-7 (Gating & robust costs)
  Residual r_k = y_k - H x_{k|k-1}；chi-square gate g_k = r_k^2 / S_k, S_k = H P_{k|k-1} H^T + R; if g_k > chi2_alpha, down-weight or reject.
  Huber loss: rho(r) = { 0.5 r^2, |r|≤c ; c(|r|-0.5 c), |r|>c } (equivalent to adaptive inflation of R).
• S606-8 (Dual-form arrival delta)
  delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |。

V. Implementation Flow M60-6 (Servo Modeling & Filtering)

1. Noise modeling & base alignment
   • Build windows on tau_mono; estimate sigma_ADEV(τ), tdev(τ); derive orders/magnitudes of Q (white PM → R; white/RW FM → Q).
   • Apply protocol compensation and log both arrival forms; compute delta_form.
2. Topology selection
   • With SyncE: FLL or low-gain PI + KF; without hardware hold: composite PLL–FLL + KF.
   • Long holdover: enable third-order [phi,f,a].
3. Gain/covariance tuning
   • Choose target B, zeta and offset_p99 budget; map to initial Kp, Ki (, Kii) via S606-6.
   • Set R from protocol measurement noise (Ch. 4/5); fit Q from sigma_ADEV; minimize overshoot/settling via replay.
4. Robust gating
   • Configure chi-square gate or Huber constant c; down-weight/reject outliers; trigger state transitions (track → holdover) on persistent gates.
   • Under observation starvation (loss/alarms): freeze updates; let state evolve by Q.
5. Clock steering & publication
   • Apply u_k to local DCO/NCO; log offset/skew/J, tdev/MTIE;
   • Persist manifest.sync.servo.* (model, parameters, gates, evaluations, signature).

VI. Contracts & Assertions

• C60-61 (Stability): Closed-loop poles satisfy |z_i| < 1 - eps; overshoot ≤ M_p; settling_time ≤ T_settle.
• C60-62 (Noise match): KF_innov_var ≈ S_k (innovation variance consistent within tol_innov).
• C60-63 (SLO gates): offset_p99 ≤ offset_budget, tdev(τ_grid) ≤ tdev_budget(τ_grid).
• C60-64 (Robustness): gated_ratio ≤ gate_max; consecutive gates beyond threshold → holdover.
• C60-65 (Dual-form consistency): delta_form ≤ tol_Tarr.
• C60-66 (Dimensional checks): check_dim(all expr) = true.

VII. Implementation Bindings I60- (Servo & Filtering APIs)*

• design_servo(noise_report, targets) -> {topology, Kp, Ki, Kii, Q, R}
• pll_pi_step(y_k, Kp, Ki, state) -> {u_k, state'}
• pll_pii_step(y_k, Kp, Ki, Kii, state) -> {u_k, state'}
• fll_step(y_k, y_km1, Kf, Ts) -> u_k
• kf_predict(A, B, x, P, Q, u) -> {x_pred, P_pred}
• kf_update(H, x_pred, P_pred, R, y, robust) -> {x_upd, P_upd, gate_stats}
• auto_gate(residual, S, policy) -> weight
• tune_from_adev(sigma_ADEV, Ts, model_order) -> {Q, R}
• Invariants: state ∈ {free, acquire, track, holdover} lawful transitions; sum(U_components^2) ≈ U_total^2; manifest is replayable.

VIII. Cross-References

• Measurement links & timestamps: Chapter 4.
• Protocol parameters & asymmetry compensation: Chapter 5.
• Allan family & time-base noise: EFT.WP.Metrology.TimeBase v1.0 Chapter 7.
• Compliance, SLOs & release: this volume Chapter 14 and Methods.Cleaning v1.0 Chapter 10.
• Statistical uncertainty & gating tests: Methods.CrossStats v1.0 Appendix E and Appendix B.

IX. Quality SLIs & Risk Control

1. SLI/SLO set: offset_p50/p95/p99, tdev@{τ}, MTIE@{τ}, settling_time, overshoot, gate_rate, holdover_time; innovation stats innov_mean≈0, innov_var/S_k ≈ 1.
2. Risk mitigation:
   • If offset_p99 breaches → reduce bandwidth B, enable FLL assist, or switch upstream.
   • Persistent gating → holdover and escalate to third-order model.
   • Long-term innov_var mismatch → re-estimate Q/R or enable robust loss.
   • Abnormal dual-form deltas → re-check asym/medium tempco and re-calibrate links.

Summary