# End-to-end verified solution for damage-robust vHS phase_lead ## What was verified The executable study is: ```text research/optimizing_pl/codex/robust_vhs_phase_lead_optimization.m ``` It was run successfully offline in MATLAB. It evaluated: ```text 918 phase_lead candidates 19 healthy/damaged/further-damaged frequency-domain cases 33 selected candidates x 13 rational full-record time-domain cases ``` Outputs: ```text robust_vhs_phase_lead_result.mat robust_vhs_phase_lead_report.md robust_vhs_phase_lead_summary.png ``` No Speedgoat or physical workflow was run. ## Main conclusion A static phase_lead alone is not the robust solution. With the damage envelope expanded beyond the two measured endpoints, the best no-authority candidate is `no lead`, and it still fails the robustness target: ```text alpha = 1 best case: no lead min NCF = 0.1846 min GM = 3.410 dB max pole = 0.999907 worst case = measured-shape rho 2, predicted k ``` The best verified end-to-end candidate is essentially an authority reduction: ```text dd_to_actuator = 0.6 * G_tiny * dd G_tiny: center = 3 Hz ratio = 1.05 zero = 2.9277 Hz pole = 3.07409 Hz ``` Its verified robust margins are: ```text min NCF = 0.3677 min GM = 7.435 dB max coupled pole = 0.999866 max full-record J4 = 72.288 % worst margin case = measured-shape rho 2, predicted k ``` Pure saved phase_leads are not robust over this envelope: ```text pl_sp1_5000, alpha 1: min NCF = 0.0514 min GM = 0.935 dB max pole = 1.002172 worst case = 1.15*damaged H, healthy k pl_sp1_7000, alpha 1: min NCF = 0.0029 min GM = 2.251 dB max pole = 1.028860 worst case = scalar H gain 2, predicted k ``` So the production answer is not "find a more aggressive phase_lead". The answer is to split the problem: ```text 1. Offline: design against an ensemble using worst-case NCF/GM/pole constraints. 2. Online: add one authority scalar a(t) that can only move toward safer command authority when damage grows. ``` ## Q1: best optimization criterion Use the coupled vHS loop for every damage case `i`: ```text A_i(z) = feedback(C_pid(z) * H_i(z), 1) L_i(z) = -N(z) * (k_i/1000) * G_T(z) * A_i(z) ``` where `N(z)` is the numerical substructure force-to-`dd` model. Hard gates: ```text max_i abs(z_pole_i) < 1 min_i NCF_i >= 0.30 absolute floor min_i NCF_i >= 0.35 preferred design floor min_i GM_i >= 7 dB max_i peak_command_ratio_i <= 1.25 ``` Then minimize: ```text score = max_i(J4_i) + 0.15*max_i(response_nrmse_i) + 0.02*max_i(tracking_nrmse_i) + margin_shortfall_penalties ``` The important change from `optimize_vHS_phase_lead_min_ncf.m` is that every margin and every response metric is computed over the damage ensemble, not one selected `specimen_file` and one selected `H_dc_dm`. ## Q2: predicting damage effect on `H_dc_dm` The measured 7000-to-5000 damage is well represented as multiplicative actuator gain plus stiffness loss: ```text H_damaged ~= 2.4247 * H_healthy complex RMS fit error over 0.1-80 Hz = 15.756 % k_damaged/k_healthy = 0.7473 ``` Use this as an uncertainty envelope: ```text gamma(rho) = 1 + rho*(2.4247 - 1) k_scale(rho) = 1 + rho*(0.7473 - 1) H_rho = gamma(rho) * H0 k_rho = k_scale(rho) * k0 ``` `rho = 0` is healthy, `rho = 1` is the measured 5000 state, and `rho > 1` is excess damage. For a different initial `H_dc_dm`, use that new healthy `H0` and apply the same multiplicative envelope. Do not put `H5000/H7000` directly into a time-domain block. Use the scalar gain model for rational time-domain simulations and use measured-shape frequency responses for margin screening. ## Q3: online stability enhancement Do not tune `P`, `I`, or phase_lead ratio live as the safety action. Those directions are not monotonic across the damage cases. Use one online authority scalar: ```text dd_to_actuator = a(t) * G_T(z) * dd ``` or, equivalently, define a safety knob `kappa = 1 - a` and increase `kappa` when the margin indicator degrades. The verified table for the best candidate shows the direction: ```text scale 1.00: min NCF 0.3677, min GM 7.435 dB scale 0.90: min NCF 0.4016, min GM 8.350 dB scale 0.70: min NCF 0.4708, min GM 10.533 dB scale 0.50: min NCF 0.5407, min GM 13.456 dB ``` ## Notch option A focused notch-filter study was added: ```text notch_filter_study.m notch_filter_study_report.md notch_filter_study_result.mat notch_filter_study_summary.png ``` The tested notch form was: ```text A(s) = (s^2 + 2*zeta_z*w0*s + w0^2)/(s^2 + 2*zeta_p*w0*s + w0^2) ``` with unity DC/high-frequency gain and attenuation at `w0` equal to `zeta_z/zeta_p`. Result: ```text No filter: min NCF = 0.1846 min GM = 3.410 dB max J4 = 88.340 % Best notch-only: center = 38 Hz depth = 0.35 zeta_pole = 0.08 min NCF = 0.2537 min GM = 7.232 dB max J4 = 89.096 % Authority 0.6 only: min NCF = 0.3834 min GM = 7.847 dB max J4 = 73.097 % Best notch + authority: alpha = 0.6 center = 42 Hz depth = 0.8 zeta_pole = 0.08 min NCF = 0.3981 min GM = 9.071 dB max J4 = 73.181 % ``` Interpretation: notch-only helps compared with no filter, but it does not meet the `NCF >= 0.35` robust target. Notch plus authority scaling gives a modest margin improvement over scalar authority alone with nearly the same performance. So the useful version is: ```text dd_to_actuator = alpha * A_notch(z) * dd ``` not notch-only. Use online OSI to decide when to move `a(t)`: ```text estimated NCF >= 0.45: slowly relax a -> nominal estimated NCF < 0.40 sustained: reduce a using a prevalidated table estimated NCF < 0.30-0.35 or resonance trip: set a = a_min and pause/ramp down stroke/force/velocity/saturation trip: ignore estimator and pause/ramp down ``` Method 7 from `research/OSI` is the right always-on monitor because it uses the closed-loop resonance in `dm` and does not require the command to excite the binding frequency. ## Production implementation path Inside the real optimizer, make these changes: ```text 1. Build a damage ensemble of (H_i, k_i), using measured endpoints and extrapolated multiplicative damage. 2. Replace single-case NCF/GM/pole metrics with min_i/max_i ensemble metrics. 3. Simulate full-record time response for the measured/scalar rational members. 4. Keep measured-shape cases frequency-domain only. 5. Add the authority scalar a as a separate online safety block, not as PID retuning. ``` The verified script in this folder is the reference implementation for those changes.