# Robust vHS phase_lead optimization report

Generated by `robust_vhs_phase_lead_optimization.m`.

## Damage model findings

- Best scalar fit: `H_damaged ~= 2.4247 * H_healthy`.
- Scalar fit complex RMS error over 0.1-80 Hz: `15.756 %`.
- Healthy stiffness: `7200643.8 N/m`.
- Damaged stiffness: `5380758.1 N/m` (`0.7473` of healthy).

## Structural modes

- Reference FEM modes: `2.422, 8.75, 18.14 Hz`.
- Numerical-substructure condensed floor modes: `2.343, 8.479, 17.77 Hz`.
- Highest condensed floor mode: `17.7728 Hz`.

## Best candidate

- Label: `alpha 0.6, lead 3 Hz x 1.05`
- Alpha: `0.6`
- Center: `3 Hz`
- Ratio: `1.05`
- Zero: `2.9277 Hz`
- Pole: `3.07409 Hz`

## Best no-authority candidate

The best candidate with `alpha = 1` is `lowpass order 1 fc 35.55 Hz (2x mode)`, with `min NCF = 0.2334`, `min GM = 7.211 dB`, and `max pole = 0.999914`. It is not feasible against the robust envelope.

No-lead full-record check: `max J4 = 88.340 %`, `min NCF = 0.1846`, `min GM = 3.410 dB`, worst case `measured-shape rho 2, predicted k`.

## Saved phase_lead checks

Only saved-lead rows forced through full-record time simulation are shown.

| Id | Label | min NCF | min GM dB | max pole | max J4 % | final score | worst NCF case |
|---:|---|---:|---:|---:|---:|---:|---|
| 905 | saved pl_sp1_5000.mat alpha 1 | 0.0514 | 0.935 | 1.002172 | 20364275371181061837265184822116897326948473318137321322239229952.000 | 21358127171192608660303715666960681229731362809034052616862040064.000 | 1.15*damaged H, healthy k |
| 911 | saved pl_sp1_5000.mat alpha 0.6 | 0.1099 | 2.014 | 0.999863 | 58.845 | 739.549 | measured-shape rho 2, predicted k |

## Low-pass candidates after the highest mode

These candidates have DC gain 1 and cut off after the highest condensed floor mode. They preserve low-frequency command authority but add lag and attenuation near/above cutoff.

| Id | Label | min NCF | min GM dB | max pole | max J4 % | max command ratio | final score | worst NCF case |
|---:|---|---:|---:|---:|---:|---:|---:|---|
| 922 | lowpass order 1 fc 35.55 Hz (2x mode) | 0.2334 | 7.211 | 0.999914 | 96.233 | 0.998 | 152.475 | healthy H, healthy k |
| 929 | lowpass order 2 fc 44.43 Hz (2.5x mode) | 0.2297 | 7.018 | 0.999916 | 97.521 | 1.000 | 156.640 | healthy H, healthy k |
| 921 | lowpass order 1 fc 26.66 Hz (1.5x mode) | 0.2264 | 7.036 | 0.999917 | 98.903 | 0.996 | 160.608 | healthy H, healthy k |
| 930 | lowpass order 2 fc 53.32 Hz (3x mode) | 0.2340 | 6.177 | 0.999914 | 95.928 | 1.000 | 165.246 | healthy H, healthy k |
| 923 | lowpass order 1 fc 44.43 Hz (2.5x mode) | 0.2378 | 5.944 | 0.999913 | 94.634 | 0.999 | 169.924 | healthy H, healthy k |
| 920 | lowpass order 1 fc 22.22 Hz (1.2x mode) | 0.2209 | 6.629 | 0.999919 | 101.037 | 0.994 | 169.968 | healthy H, healthy k |
| 928 | lowpass order 2 fc 35.55 Hz (2x mode) | 0.2232 | 6.238 | 0.999918 | 99.958 | 1.000 | 175.834 | healthy H, healthy k |
| 919 | lowpass order 1 fc 19.55 Hz (1.1x mode) | 0.2166 | 6.393 | 0.999920 | 102.775 | 0.993 | 180.019 | healthy H, healthy k |
| 924 | lowpass order 1 fc 53.32 Hz (3x mode) | 0.2407 | 5.129 | 0.999912 | 93.573 | 0.999 | 214.522 | healthy H, healthy k |
| 927 | lowpass order 2 fc 26.66 Hz (1.5x mode) | 0.2124 | 5.248 | 0.999921 | 104.148 | 1.000 | 239.025 | healthy H, healthy k |
| 926 | lowpass order 2 fc 22.22 Hz (1.2x mode) | 0.2038 | 4.672 | 0.999924 | 107.615 | 1.000 | 297.284 | healthy H, healthy k |
| 925 | lowpass order 2 fc 19.55 Hz (1.1x mode) | 0.1969 | 4.330 | 0.999927 | 110.526 | 1.000 | 341.110 | healthy H, healthy k |

## Top candidates

| Id | Label | min NCF | min GM dB | max pole | max J4 % | final score | worst NCF case |
|---:|---|---:|---:|---:|---:|---:|---|
| 816 | alpha 0.6, lead 3 Hz x 1.05 | 0.3677 | 7.435 | 0.999866 | 72.288 | 84.333 | measured-shape rho 2, predicted k |
| 826 | alpha 0.6, lead 4 Hz x 1.05 | 0.3678 | 7.440 | 0.999866 | 72.306 | 84.352 | measured-shape rho 2, predicted k |
| 806 | alpha 0.6, lead 2 Hz x 1.05 | 0.3675 | 7.431 | 0.999867 | 72.420 | 84.486 | measured-shape rho 2, predicted k |
| 836 | alpha 0.6, lead 6 Hz x 1.05 | 0.3682 | 7.451 | 0.999866 | 72.432 | 84.495 | measured-shape rho 2, predicted k |
| 846 | alpha 0.6, lead 8 Hz x 1.05 | 0.3686 | 7.464 | 0.999866 | 72.547 | 84.625 | measured-shape rho 2, predicted k |
| 796 | alpha 0.6, lead 1.5 Hz x 1.05 | 0.3675 | 7.429 | 0.999867 | 72.612 | 84.707 | measured-shape rho 2, predicted k |
| 856 | alpha 0.6, lead 12 Hz x 1.05 | 0.3696 | 7.493 | 0.999866 | 72.699 | 84.797 | measured-shape rho 2, predicted k |
| 898 | alpha 0.6, lead 60 Hz x 1.2 | 0.3706 | 7.598 | 0.999866 | 72.770 | 84.875 | measured-shape rho 2, predicted k |
| 866 | alpha 0.6, lead 18 Hz x 1.05 | 0.3713 | 7.542 | 0.999866 | 72.820 | 84.931 | measured-shape rho 2, predicted k |
| 887 | alpha 0.6, lead 40 Hz x 1.1 | 0.3708 | 7.561 | 0.999866 | 72.843 | 84.959 | measured-shape rho 2, predicted k |
| 876 | alpha 0.6, lead 27 Hz x 1.05 | 0.3739 | 7.615 | 0.999866 | 72.907 | 85.026 | measured-shape rho 2, predicted k |
| 897 | alpha 0.6, lead 60 Hz x 1.1 | 0.3767 | 7.711 | 0.999866 | 72.926 | 85.042 | measured-shape rho 2, predicted k |
| 886 | alpha 0.6, lead 40 Hz x 1.05 | 0.3770 | 7.699 | 0.999866 | 72.967 | 85.088 | measured-shape rho 2, predicted k |
| 896 | alpha 0.6, lead 60 Hz x 1.05 | 0.3799 | 7.776 | 0.999866 | 73.010 | 85.131 | measured-shape rho 2, predicted k |
| 774 | attenuation 0.6 | 0.3834 | 7.847 | 0.999866 | 73.097 | 85.224 | measured-shape rho 2, predicted k |

## Safety attenuation check for best candidate

| Scale | min NCF | min GM dB | worst NCF case |
|---:|---:|---:|---|
| 1.000 | 0.3677 | 7.435 | measured-shape rho 2, predicted k |
| 0.950 | 0.3846 | 7.881 | measured-shape rho 2, predicted k |
| 0.900 | 0.4016 | 8.350 | measured-shape rho 2, predicted k |
| 0.850 | 0.4188 | 8.847 | measured-shape rho 2, predicted k |
| 0.800 | 0.4360 | 9.373 | measured-shape rho 2, predicted k |
| 0.750 | 0.4534 | 9.934 | measured-shape rho 2, predicted k |
| 0.700 | 0.4708 | 10.533 | measured-shape rho 2, predicted k |
| 0.650 | 0.4883 | 11.177 | measured-shape rho 2, predicted k |
| 0.600 | 0.5058 | 11.872 | measured-shape rho 2, predicted k |
| 0.550 | 0.5233 | 12.628 | measured-shape rho 2, predicted k |
| 0.500 | 0.5407 | 13.456 | measured-shape rho 2, predicted k |
| 0.450 | 0.5581 | 14.371 | measured-shape rho 2, predicted k |
| 0.400 | 0.5754 | 15.394 | measured-shape rho 2, predicted k |
| 0.350 | 0.5919 | 16.554 | damaged H, healthy k |
| 0.300 | 0.6082 | 17.893 | damaged H, healthy k |
| 0.250 | 0.6247 | 19.476 | damaged H, healthy k |
| 0.200 | 0.6413 | 21.415 | damaged H, healthy k |
| 0.150 | 0.6580 | 23.913 | damaged H, healthy k |
| 0.100 | 0.6745 | 27.435 | damaged H, healthy k |
| 0.050 | 0.6910 | 33.456 | damaged H, healthy k |

## Interpretation

The optimizer ranks candidates by worst-case margin first and then by full-record time-domain error on the measured/scalar rational cases. A static phase_lead is accepted only if it survives the healthy, damaged, mismatch, scalar-gain, and measured-shape frequency-response envelope.

The safety table is not assumed monotonic by theory. It is an offline verification table for using command attenuation as the single online safety knob.